real analysis - Difference between boundary point & limit point.

<!DOCTYPE html> <html itemscope itemtype="https://schema.org/QAPage" class="html__responsive " lang="en"> <head> <title>real analysis - Difference between boundary point & limit point. - Mathematics Stack Exchange</title> <link rel="shortcut icon" href="https://cdn.sstatic.net/Sites/math/Img/favicon.ico?v=92addaa54d18"> <link rel="apple-touch-icon" href="https://cdn.sstatic.net/Sites/math/Img/apple-touch-icon.png?v=0ae50baa40ed"> <link rel="image_src" href="https://cdn.sstatic.net/Sites/math/Img/apple-touch-icon.png?v=0ae50baa40ed"> <link rel="search" type="application/opensearchdescription+xml" title="Mathematics Stack Exchange" href="/opensearch.xml"> <link rel="canonical" href="https://math.stackexchange.com/questions/1290529/difference-between-boundary-point-limit-point" /> <meta name="viewport" content="width=device-width, height=device-height, initial-scale=1.0, minimum-scale=1.0"> <meta property="og:type" content= "website" /> <meta property="og:url" content="https://math.stackexchange.com/questions/1290529/difference-between-boundary-point-limit-point"/> <meta property="og:site_name" content="Mathematics Stack Exchange" /> <meta property="og:image" itemprop="image primaryImageOfPage" content="https://cdn.sstatic.net/Sites/math/Img/apple-touch-icon@2.png?v=4ec1df2e49b1" /> <meta name="twitter:card" content="summary"/> <meta name="twitter:domain" content="math.stackexchange.com"/> <meta name="twitter:title" property="og:title" itemprop="name" content="Difference between boundary point & limit point." /> <meta name="twitter:description" property="og:description" itemprop="description" content="A limit point is just a accumulation point whose neighbourhood contains infinitely many elements of the sequence. 

Is there any difference between boundary point &amp; limit point? I've read in an..." /> <script id="webpack-public-path" type="text/uri-list">https://cdn.sstatic.net/</script> <script src="https://ajax.googleapis.com/ajax/libs/jquery/3.7.1/jquery.min.js"></script> <script defer src="https://cdn.sstatic.net/Js/third-party/npm/@stackoverflow/stacks/dist/js/stacks.min.js?v=90a447add688"></script> <script src="https://cdn.sstatic.net/Js/stub.en.js?v=30464efc364f"></script> <link rel="stylesheet" type="text/css" href="https://cdn.sstatic.net/Shared/stacks.css?v=bd2aba62c5a4"> <link rel="stylesheet" type="text/css" href="https://cdn.sstatic.net/Sites/math/primary.css?v=6953e6c83224"> <link rel="alternate" type="application/atom+xml" title="Feed for question 'Difference between boundary point & limit point.'" href="/feeds/question/1290529"> <script> StackExchange.ready(function () { StackExchange.using("postValidation", function () { StackExchange.postValidation.initOnBlurAndSubmit($('#post-form'), 2, 'answer'); }); StackExchange.question.init({showCitation:true,showAnswerHelp:true,totalCommentCount:5,shownCommentCount:5,enableTables:true,questionId:1290529}); styleCode(); StackExchange.realtime.subscribeToQuestion('69', '1290529'); StackExchange.using("gps", function () { StackExchange.gps.trackOutboundClicks('#content', '.js-post-body'); }); }); </script> <link rel="stylesheet" type="text/css" href="https://cdn.sstatic.net/Shared/Channels/channels.css?v=5981bb1a5bd7"> <script src="https://cdn.sstatic.net/Js/third-party/citation-helper.js?v=a4ec380e29ba"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: ["STIX", "TeX"], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { inlineMath: [["$", "$"], ["\\\$","\\\$"]], displayMath: [["$$", "$$"], ["\\[", "\\]"]], processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { extensions: ["begingroup.js"], noUndefined: { attributes: { mathcolor: "red", mathbackground: "#FFEEEE", mathsize: "90%" } }, Macros: { href: "{}" } }, messageStyle: "none", styles: { ".MathJax_Display, .MathJax_Preview, .MathJax_Preview > *": { "background": "inherit" } }, SEEditor: "mathjaxEditing" }); </script> <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-AMS_HTML-full"></script> <script> StackExchange.ready(function () { StackExchange.realtime.init('wss://qa.sockets.stackexchange.com'); StackExchange.realtime.subscribeToReputationNotifications('69'); StackExchange.realtime.subscribeToTopBarNotifications('69'); }); </script> <script type="application/json" data-role="module-args" data-module-name="Shared/options.mod">{"options":{"locale":"en","serverTime":1740539661,"routeName":"Questions/Show","stackAuthUrl":"https://stackauth.com","networkMetaHostname":"meta.stackexchange.com","site":{"name":"Mathematics Stack Exchange","description":"Q\u0026A for people studying math at any level and professionals in related fields","isNoticesTabEnabled":true,"enableNewTagCreationWarning":true,"insertSpaceAfterNameTabCompletion":false,"id":69,"cookieDomain":".stackexchange.com","childUrl":"https://math.meta.stackexchange.com","negativeVoteScoreFloor":null,"enableSocialMediaInSharePopup":true,"protocol":"https"},"user":{"fkey":"b39cb1a761f93912de068f8e6267ee5702b7d597a570755c5112354b099addd4","tid":"44a4f5b0-7991-4597-b582-6dd071c0ba8d","rep":0,"isAnonymous":true,"isAnonymousNetworkWide":true},"realtime":{"newest":true,"active":true,"tagged":true,"staleDisconnectIntervalInHours":0},"events":{"postType":{"question":1},"postEditionSection":{"title":1,"body":2,"tags":3}}}}</script> <script type="application/json" data-role="module-args" data-module-name="Shared/settings.mod">{"settings":{"questionLinkTitleReplacement":{"maxNumberOfSitesProcessed":10,"maxReplacementsPerSite":20},"comments":{},"elections":{"opaVoteResultsBaseUrl":"https://www.opavote.com/results/"},"markdown":{"enableTables":true},"paths":{"jQueryUIJSPath":"https://ajax.googleapis.com/ajax/libs/jqueryui/1.12.0/jquery-ui.min.js","jQueryUICSSPath":"https://ajax.googleapis.com/ajax/libs/jqueryui/1.12.0/themes/smoothness/jquery-ui.css"},"site":{"enableUserHovercards":true,"stacksEditorPreviewEnabled":true,"forceHttpsImages":true,"allowImageUploads":true,"enableImageHttps":true},"auth":{"oauthInPopup":true},"mentions":{"maxNumUsersInDropdown":50},"flags":{"allowRetractingCommentFlags":true,"allowRetractingFlags":true},"search":{},"intercom":{"appId":"inf0secd"},"snippets":{"renderDomain":"stacksnippets.net"},"userMessaging":{},"questions":{"enableQuestionTitleLengthLiveWarning":true,"questionTitleLengthStartLiveWarningChars":50,"maxTitleSize":150,"enableSavesFeature":true},"tags":{},"legal":{"useCustomConsent":false,"oneTrustTCFConfigId":"cb0f3c87-b769-4e66-bbaa-377f9194216d"},"subscriptions":{"defaultMaxTrueUpSeats":1000,"defaultFreemiumMaxTrueUpSeats":50,"defaultBasicMaxTrueUpSeats":250},"accounts":{"currentPasswordRequiredForChangingStackIdPassword":true},"image":{"maxImageUploadSizeInBytesAnimatedGif":2097152,"maxImageUploadSizeInBytes":10485760}}}</script> <script>StackExchange.init();</script> <script> StackExchange.using.setCacheBreakers({"Js/adops.en.js":"6da43f5e0a84","Js/ask.en.js":"","Js/begin-edit-event.en.js":"20edbaccceae","Js/copy-transpiled.en.js":"f48f8f039185","Js/events.en.js":"","Js/explore-qlist.en.js":"ee2a4f8c3992","Js/full-anon.en.js":"427705904ec4","Js/full.en.js":"21465ad85806","Js/highlightjs-loader.en.js":"a3e36fe85053","Js/inline-tag-editing.en.js":"8517756a2cb6","Js/keyboard-shortcuts.en.js":"c255a5a5979b","Js/markdown-it-loader.en.js":"781aa77e9602","Js/mentions-transpiled.en.js":"bcadb48d3570","Js/moderator.en.js":"e891f78ea17e","Js/postCollections-transpiled.en.js":"9f7a87e8cf0d","Js/post-validation.en.js":"44d59ccfc490","Js/question-editor.en.js":"","Js/review-v2-transpiled.en.js":"c9e05881f8ab","Js/revisions.en.js":"9dd135bb585f","Js/stacks-editor.en.js":"cf77c9d2187c","Js/tageditor.en.js":"5b8f289d02ef","Js/tageditornew.en.js":"ca426894aed0","Js/tagsuggestions.en.js":"762b96a1ae1f","Js/unlimited-transpiled.en.js":"8713a979101d","Js/wmd.en.js":"eef1c2cfa021","Js/mathjax-editing.v3.en.js":"4f38b81017dc","Js/mathjax-editing.en.js":"2581d730d948"}); StackExchange.using("gps", function() { StackExchange.gps.init(false); }); </script> <noscript id="noscript-css"><style>body,.s-topbar{margin-top:1.9em}</style></noscript> </head> <body class="question-page unified-theme"> <div id="signup-modal-container"></div> <script type="application/json" data-role="module-args" data-module-name="islands/signup-modal/index.mod">{"ContainerElementId":"signup-modal-container","FKey":"b39cb1a761f93912de068f8e6267ee5702b7d597a570755c5112354b099addd4","TriggerEvent":"signupModalShow","OauthInPopup":true,"ReturnUrl":"https://math.stackexchange.com/questions/1290529/difference-between-boundary-point-limit-point/1290541","ReturnUrlForPopup":"https://math.stackexchange.com/users/after-signup/oauth-only","SiteName":"Mathematics","SiteLogoPath":"https://cdn.sstatic.net/Sites/math/Img/icon-48.png?v=78e7528776aa","AuthProviders":["Google"],"ParentSiteUrl":"","IsInitiallyVisible":false}</script> <script defer src="https://cdn.sstatic.net/Js/webpack-chunks/svelte.en.js?v=f1f216c36f5e"></script><script defer src="https://cdn.sstatic.net/Js/webpack-chunks/stacks-svelte.en.js?v=e5a0c3f4bb1b"></script><script defer src="https://cdn.sstatic.net/Js/webpack-chunks/8901.en.js?v=9ff34567d3a5"></script><script defer src="https://cdn.sstatic.net/Js/islands/signup-modal.en.js?v=47ec828efb46"></script> <script defer> dispatchEvent(new CustomEvent("openSignupModal")); </script> <div id="one-tap-container"></div> <script type="application/json" data-role="module-args" data-module-name="islands/one-tap/index.mod">{"ContainerElementId":"one-tap-container","FKey":"b39cb1a761f93912de068f8e6267ee5702b7d597a570755c5112354b099addd4","GoogleClientId":"717762328687-iludtf96g1hinl76e4lc1b9a82g457nn.apps.googleusercontent.com","Autoselect":false,"ReturnUrl":"https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1290529%2fdifference-between-boundary-point-limit-point%2f1290541"}</script><script defer src="https://cdn.sstatic.net/Js/webpack-chunks/svelte.en.js?v=f1f216c36f5e"></script><script defer src="https://cdn.sstatic.net/Js/islands/one-tap.en.js?v=bc67e8c0f01e"></script> <div id="notify-container"></div> <div id="custom-header"></div> <header class="s-topbar ps-fixed t0 l0 js-top-bar"> <a href="#content" class="s-topbar--skip-link">Skip to main content</a> <div class="s-topbar--container"> <a href="#" class="s-topbar--menu-btn js-left-sidebar-toggle" role="menuitem" aria-haspopup="true" aria-controls="left-sidebar" aria-expanded="false"><span></span></a> <div class="topbar-dialog leftnav-dialog js-leftnav-dialog dno"> <div class="left-sidebar js-unpinned-left-sidebar" data-can-be="left-sidebar" data-is-here-when="sm"></div> </div> <a href="#" class="s-topbar--logo network-logo js-gps-track js-network-logo" data-gps-track="stack_exchange_popup.show" role="menuitem" aria-haspopup="true" aria-controls="topbar-network-logo-dialog" aria-expanded="false"> <svg aria-hidden="true" class="native mtn1 svg-icon iconSEAlternativeSm" width="107" height="15" viewBox="0 0 107 15"><path fill="#FEFEFE" d="m48.41 11.93-1.96-3.2-1.04 1.16v2.04h-1.42V2.18h1.42v6.01L48.14 5h1.72l-2.44 2.7 2.74 4.22zm-7.06.08c-1.59 0-3.14-.96-3.14-3.56s1.55-3.54 3.14-3.54c.97 0 1.65.27 2.31.97l-.97.93c-.44-.48-.79-.66-1.34-.66q-.84.01-1.3.62c-.31.38-.42.87-.42 1.68s.1 1.32.41 1.7c.3.4.76.62 1.3.62.56 0 .9-.18 1.35-.66l.97.92c-.66.7-1.34.98-2.31.98m-5.66-3.15h-1.65c-.83 0-1.26.37-1.26 1s.4.99 1.3.99c.53 0 .93-.04 1.3-.4q.32-.29.31-1.03zm.03 3.07v-.63c-.51.5-1 .71-1.87.71s-1.46-.2-1.89-.63a2 2 0 0 1-.55-1.49c0-1.16.82-2 2.42-2h1.86v-.5c0-.87-.44-1.3-1.54-1.3-.77 0-1.15.18-1.54.68l-.92-.86c.66-.77 1.35-1 2.52-1q2.9 0 2.9 2.38v4.64zm-5.9 0c-1.32 0-1.93-.93-1.93-1.93V6.18h-.8V5.1h.8V3h1.41v2.1h1.36v1.07H29.3v3.75c0 .5.25.81.78.81h.58v1.2zm-6.33.08c-1.48 0-2.55-.34-3.49-1.28l1-.98c.72.72 1.51.94 2.52.94 1.3 0 2.04-.55 2.04-1.5q0-.65-.39-1.01c-.25-.23-.5-.33-1.08-.41l-1.16-.17a3.4 3.4 0 0 1-1.88-.78 2.4 2.4 0 0 1-.72-1.86c0-1.7 1.25-2.86 3.3-2.86 1.3 0 2.22.33 3.07 1.1l-.96.94a3 3 0 0 0-2.15-.75c-1.16 0-1.8.65-1.8 1.52q-.02.54.37.9c.25.22.65.38 1.11.45l1.13.17c.91.13 1.42.35 1.84.72.54.47.8 1.17.8 2 0 1.8-1.48 2.86-3.55 2.86"/><path fill="#2F96E8" d="M104.16 7.09c-.2-.42-.6-.74-1.2-.74s-.99.32-1.18.74c-.1.25-.15.44-.16.75h2.7a2 2 0 0 0-.16-.75m-2.54 1.96c0 .9.56 1.57 1.55 1.57.78 0 1.16-.21 1.61-.66l1.08 1.04a3.4 3.4 0 0 1-2.7 1.11c-1.68 0-3.29-.76-3.29-3.62 0-2.3 1.26-3.6 3.1-3.6 1.97 0 3.1 1.44 3.1 3.37v.79zm-5.48-2.57C95.1 6.48 95 7.37 95 8.3s.1 1.85 1.15 1.85 1.18-.91 1.18-1.85c0-.93-.13-1.82-1.18-1.82m-.17 8.22c-1.1 0-1.84-.21-2.58-.92l1.1-1.11c.4.38.8.54 1.4.54 1.06 0 1.43-.74 1.43-1.46v-.72c-.47.51-1 .7-1.7.7-.69 0-1.29-.23-1.68-.62-.67-.66-.73-1.57-.73-2.8 0-1.24.06-2.13.73-2.8.4-.39 1-.62 1.7-.62.75 0 1.24.2 1.73.75v-.67h1.72v6.8c0 1.7-1.21 2.93-3.12 2.93m-5.76-2.67V7.76c0-.96-.61-1.28-1.17-1.28s-1.18.32-1.18 1.28v4.27h-1.78V4.97h1.73v.65a2.4 2.4 0 0 1 1.78-.73q1.07.02 1.67.62c.58.57.73 1.24.73 2v4.52zm-7.1-2.98h-1.4c-.64 0-1 .3-1 .8 0 .49.33.81 1.02.81.5 0 .8-.04 1.12-.34q.28-.25.26-.89zm.04 2.98v-.6c-.48.47-.93.67-1.74.67q-1.2 0-1.82-.62c-.38-.4-.58-.97-.58-1.59 0-1.12.77-2.05 2.42-2.05h1.68V7.5c0-.77-.38-1.11-1.32-1.11-.68 0-1 .16-1.37.58l-1.13-1.1c.7-.75 1.38-.97 2.57-.97q3 0 3.02 2.5v4.64zm-6.93 0v-4.3c0-.94-.6-1.25-1.15-1.25-.56 0-1.15.32-1.15 1.24v4.31h-1.77V2.38h1.77v3.24a2.4 2.4 0 0 1 1.7-.73c1.56 0 2.38 1.08 2.38 2.57v4.57zm-6.96.08c-1.42 0-3.18-.76-3.18-3.62 0-2.85 1.76-3.6 3.18-3.6.98 0 1.72.3 2.34.95l-1.2 1.2c-.36-.4-.68-.56-1.14-.56q-.62-.01-1.01.46c-.27.33-.4.8-.4 1.55s.13 1.24.4 1.58q.39.46 1 .46c.47 0 .79-.16 1.15-.56l1.2 1.18c-.62.65-1.36.96-2.34.96m-5.53-.08-1.3-2.11-1.3 2.11H59l2.45-3.6-2.35-3.46h2.12L62.42 7l1.21-2.02h2.13L63.4 8.43l2.46 3.6zm-11.75 0V2.06h6.6V3.8h-4.65v2.33h3.96v1.74h-3.96v2.42h4.65v1.74z"/><path fill="#8FD8F7" d="M0 3c0-1.1.9-2 2-2h8a2 2 0 0 1 2 2z"/><path fill="#155397" d="M12 10H0c0 1.1.9 2 2 2h5v3l3-3a2 2 0 0 0 2-2"/><path fill="#46A2D9" d="M0 4h12v2H0z"/><path fill="#2D6DB5" d="M0 7h12v2H0z"/></svg> </a> <div class="topbar-dialog network-logo-dialog js-network-logo-dialog dno" id="topbar-network-logo-dialog" role="dialog" aria-labelledby="topbar-network-logo-dialog-title" aria-describedby="topbar-network-logo-dialog-body"> <div class="dialog-content"> <h4 class="bold" id="topbar-network-logo-dialog-title">Stack Exchange Network</h4> <p id="topbar-network-logo-dialog-body"> Stack Exchange network consists of 183 Q&A communities including <a href="https://stackoverflow.com">Stack Overflow</a>, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. </p> <a class="s-btn s-btn__filled" href="https://stackexchange.com" data-gps-track="stack_exchange_popup.click">Visit Stack Exchange</a> <button class="s-btn s-btn__muted p0 ps-absolute t16 r16 js-close-button" aria-label="Close"><svg aria-hidden="true" class="svg-icon iconClear" width="18" height="18" viewBox="0 0 18 18"><path d="M15 4.41 13.59 3 9 7.59 4.41 3 3 4.41 7.59 9 3 13.59 4.41 15 9 10.41 13.59 15 15 13.59 10.41 9z"/></svg></button> </div> </div> <form id="search" role="search" action=/search class="s-topbar--searchbar js-searchbar " autocomplete="off"> <div class="s-topbar--searchbar--input-group"> <input name="q" type="text" role="combobox" placeholder="Search on Mathematics…" value="" autocomplete="off" maxlength="240" class="s-input s-input__search js-search-field wmn1 " aria-label="Search" aria-controls="top-search" data-controller="s-popover" data-action="focus->s-popover#show" data-s-popover-placement="bottom-start" /> <svg aria-hidden="true" class="s-input-icon s-input-icon__search svg-icon iconSearch" width="18" height="18" viewBox="0 0 18 18"><path d="m18 16.5-5.14-5.18h-.35a7 7 0 1 0-1.19 1.19v.35L16.5 18zM12 7A5 5 0 1 1 2 7a5 5 0 0 1 10 0"/></svg> <div class="s-popover p0 wmx100 wmn4 sm:wmn-initial js-top-search-popover" id="top-search" role="menu"> <div class="s-popover--arrow"></div> <div class="s-popover--content"> <div class="js-spinner p24 d-flex ai-center jc-center d-none"> <div class="s-spinner s-spinner__sm fc-orange-400"> <div class="v-visible-sr">Loading…</div> </div> </div> <span class="v-visible-sr js-screen-reader-info"></span> <div class="js-ac-results overflow-y-auto hmx3 d-none"></div> <div class="js-search-hints" aria-describedby="Tips for searching"></div> </div> </div> </div> </form> <nav class="h100 ml-auto overflow-x-auto pr12" aria-label="Topbar"> <ol class="s-topbar--content" role="menubar"> <li role="none"> <a href="/help" class="s-topbar--item js-help-button" role="menuitem" title="Help Center and other resources" aria-haspopup="true" aria-controls="topbar-help-dialog" data-ga="["top navigation","help menu click",null,null,null]"><svg aria-hidden="true" class="svg-icon iconHelp" width="18" height="18" viewBox="0 0 18 18"><path d="M9 1C4.64 1 1 4.64 1 9s3.64 8 8 8 8-3.64 8-8-3.64-8-8-8m.81 12.13c-.02.71-.55 1.15-1.24 1.13-.66-.02-1.17-.49-1.15-1.2.02-.72.56-1.18 1.22-1.16.7.03 1.2.51 1.17 1.23M11.77 8c-.59.66-1.78 1.09-2.05 1.97a4 4 0 0 0-.09.75c0 .05-.03.16-.18.16H7.88c-.16 0-.18-.1-.18-.15.06-1.35.66-2.2 1.83-2.88.39-.29.7-.75.7-1.24.01-1.24-1.64-1.82-2.35-.72-.21.33-.18.73-.18 1.1H5.75c0-1.97 1.03-3.26 3.03-3.26 1.75 0 3.47.87 3.47 2.83 0 .57-.2 1.05-.48 1.44"/></svg></a> </li> <div class="topbar-dialog help-dialog js-help-dialog dno" id="topbar-help-dialog" role="menu"> <div class="modal-content"> <ul> <li> <a href="/tour" class="js-gps-track s-block-link" data-gps-track="help_popup.click({ item_type:1 })" data-ga="["top navigation","tour submenu click",null,null,null]"> Tour <span class="item-summary"> Start here for a quick overview of the site </span> </a> </li> <li> <a href="/help" class="js-gps-track s-block-link" data-gps-track="help_popup.click({ item_type:4 })" data-ga="["top navigation","help center",null,null,null]"> Help Center <span class="item-summary"> Detailed answers to any questions you might have </span> </a> </li> <li> <a href="https://math.meta.stackexchange.com" class="js-gps-track s-block-link" data-gps-track="help_popup.click({ item_type:2 })" data-ga="["top navigation","meta submenu click",null,null,null]"> Meta <span class="item-summary"> Discuss the workings and policies of this site </span> </a> </li> <li> <a href="https://stackoverflow.co/" class="js-gps-track s-block-link" data-gps-track="help_popup.click({ item_type:6 })" data-ga="["top navigation","about us submenu click",null,null,null]"> About Us <span class="item-summary"> Learn more about Stack Overflow the company, and our products </span> </a> </li> </ul> </div> </div> <li role="none"> <a href="https://stackexchange.com" class="s-topbar--item js-site-switcher-button js-gps-track" data-gps-track="site_switcher.show" aria-label="Site switcher" role="menuitem" title="A list of all 183 Stack Exchange sites" aria-haspopup="true" aria-expanded="false" data-ga="["top navigation","stack exchange click",null,null,null]"> <svg aria-hidden="true" class="svg-icon iconStackExchange" width="18" height="18" viewBox="0 0 18 18"><path d="M15 1H3a2 2 0 0 0-2 2v2h16V3a2 2 0 0 0-2-2M1 13c0 1.1.9 2 2 2h8v3l3-3h1a2 2 0 0 0 2-2v-2H1zm16-7H1v4h16z"/></svg> </a> </li> <li class="js-topbar-dialog-corral" role="presentation"> <div class="topbar-dialog siteSwitcher-dialog dno" role="menu"> <div class="header fw-wrap"> <h3 class="flex--item"> <a href="https://math.stackexchange.com">current community</a> </h3> <div class="flex--item fl1"> <div class="ai-center d-flex jc-end"> <button class="js-close-button s-btn s-btn__muted p0 ml8 d-none sm:d-block" type="button" aria-label="Close" > <svg aria-hidden="true" class="svg-icon iconClear" width="18" height="18" viewBox="0 0 18 18"><path d="M15 4.41 13.59 3 9 7.59 4.41 3 3 4.41 7.59 9 3 13.59 4.41 15 9 10.41 13.59 15 15 13.59 10.41 9z"/></svg> </button> </div> </div> </div> <div class="modal-content bg-blue-200 current-site-container"> <ul class="current-site"> <li class="d-flex"> <div class="fl1"> <a href="https://math.stackexchange.com" class="current-site-link d-flex gx8 site-link js-gps-track" data-id="69" data-gps-track="site_switcher.click({ item_type:3 })"> <div class="favicon favicon-math site-icon flex--item" title="Mathematics"></div> <span class="flex--item fl1"> Mathematics </span> </a> </div> <div class="related-links"> <a href="https://math.stackexchange.com/help" class="js-gps-track" data-gps-track="site_switcher.click({ item_type:14 })">help</a> <a href="https://chat.stackexchange.com?tab=site&host=math.stackexchange.com" class="js-gps-track" data-gps-track="site_switcher.click({ item_type:6 })">chat</a> </div> </li> <li class="related-site d-flex"> <div class="L-shaped-icon-container"> <span class="L-shaped-icon"></span> </div> <a href="https://math.meta.stackexchange.com" class="s-block-link px16 d-flex gx8 site-link js-gps-track" data-id="70" data-gps-track="site.switch({ target_site:70, item_type:3 }),site_switcher.click({ item_type:4 })"> <div class="favicon favicon-mathmeta site-icon flex--item" title="Mathematics Meta"></div> <span class="flex--item fl1"> Mathematics Meta </span> </a> </li> </ul> </div> <div class="header" id="your-communities-header"> <h3> your communities </h3> </div> <div class="modal-content" id="your-communities-section"> <div class="call-to-login"> <a href="https://math.stackexchange.com/users/signup?ssrc=site_switcher&returnurl=https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1290529%2fdifference-between-boundary-point-limit-point%2f1290541" class="login-link js-gps-track" data-gps-track="site_switcher.click({ item_type:10 })">Sign up</a> or <a href="https://math.stackexchange.com/users/login?ssrc=site_switcher&returnurl=https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1290529%2fdifference-between-boundary-point-limit-point%2f1290541" class="login-link js-gps-track" data-gps-track="site_switcher.click({ item_type:11 })">log in</a> to customize your list. </div> </div> <div class="header"> <h3><a href="https://stackexchange.com/sites">more stack exchange communities</a> </h3> <a href="https://stackoverflow.blog" class="float-right">company blog</a> </div> <div class="modal-content"> <div class="child-content"></div> </div> </div> </li> <li role="none"><button class="s-topbar--item s-btn s-btn__icon s-btn__muted d-none sm:d-inline-flex js-searchbar-trigger" role="menuitem" aria-label="Search" aria-haspopup="true" aria-controls="search" title="Click to show search"><svg aria-hidden="true" class="svg-icon iconSearch" width="18" height="18" viewBox="0 0 18 18"><path d="m18 16.5-5.14-5.18h-.35a7 7 0 1 0-1.19 1.19v.35L16.5 18zM12 7A5 5 0 1 1 2 7a5 5 0 0 1 10 0"/></svg></button></li> <li role="none"> <a href="https://math.stackexchange.com/users/login?ssrc=head&returnurl=https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1290529%2fdifference-between-boundary-point-limit-point%2f1290541" class="s-topbar--item s-topbar--item__unset s-btn s-btn__outlined ws-nowrap js-gps-track" role="menuitem" rel="nofollow" data-gps-track="login.click" data-ga="["top navigation","login button click",null,null,null]">Log in</a> </li> <li role="none"><a href="https://math.stackexchange.com/users/signup?ssrc=head&returnurl=https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1290529%2fdifference-between-boundary-point-limit-point%2f1290541" class="s-topbar--item s-topbar--item__unset ml4 s-btn s-btn__filled ws-nowrap js-signup-button js-gps-track" role="menuitem" rel="nofollow" data-gps-track="signup.topbar.click" data-ga="["sign up","Sign Up Navigation","Header",null,null]">Sign up</a></li> </ol> </nav> </div> </header> <script> StackExchange.ready(function () { StackExchange.topbar.init(); }); StackExchange.scrollPadding.setPaddingTop(50, 10); </script> <div id="announcement-banner" class="js-announcement-banner bg-black-500 fc-white baw0 ff-sans fs-body2 py2" data-campaign="2025-02-24.community-ama" data-cookie="notice-ama2" data-expire-date="1741291200000"> <div class="d-flex jc-space-between ai-center wmx12 mx-auto px16 py8"> <div class="flex--item mr12 "> <span> Ask questions and share your thoughts on the future of Stack Overflow. Join our first live community AMA this Wednesday, February 26th, at 3 PM ET. </span> <a target="_blank" class="s-link s-link__underlined fw-bold fc-white js-link js-gps-track" href="https://meta.stackexchange.com/questions/406399/join-us-for-our-first-community-wide-ama-ask-me-anything-with-stack-overflow-s?utm_medium=ppc&utm_source=stackoverflow-community&utm_campaign=community-ama&utm_content=announcement-banner2" data-ga="["community-ama","Announcement Banner","https://meta.stackexchange.com/questions/406399/join-us-for-our-first-community-wide-ama-ask-me-anything-with-stack-overflow-s?utm_medium=ppc&utm_source=stackoverflow-community&utm_campaign=community-ama&utm_content=announcement-banner2",null,{"dimension4":"community-ama"}]" data-gps-track="announcement_banner.click({campaign: '2025-02-24.community-ama', location: 2, action: 2})">Learn more</a> </div> <a class="flex--item fc-white js-dismiss js-gps-track" href="#" title="dismiss" data-gps-track="announcement_banner.click({campaign: '2025-02-24.community-ama', location: 2, action: 1})"><svg aria-hidden="true" class="m0 svg-icon iconClear" width="18" height="18" viewBox="0 0 18 18"><path d="M15 4.41 13.59 3 9 7.59 4.41 3 3 4.41 7.59 9 3 13.59 4.41 15 9 10.41 13.59 15 15 13.59 10.41 9z"/></svg></a> </div> </div> <script> StackExchange.ready(function () { StackExchange.Notice.announcementBannerInit(); }); </script> <header class="site-header"> <div class="site-header--container"> <a class="site-header--link d-flex ai-center fs-headline1 fw-bold" href="https://math.stackexchange.com"> <img class="h-auto wmx100" src="https://cdn.sstatic.net/Sites/math/Img/logo.svg?v=4a36350d1199" alt="Mathematics"> </a> </div> </header> <div class="container"> <div id="left-sidebar" data-is-here-when="md lg" class="left-sidebar js-pinned-left-sidebar ps-relative"> <div class="left-sidebar--sticky-container js-sticky-leftnav"> <nav aria-label="Primary"> <ol class="nav-links"> <li> <ol class="nav-links"> <li class="ps-relative" aria-current="false"> <a href="/" class="s-block-link pl8 js-gps-track nav-links--link -link__with-icon" data-gps-track="top_nav.click({is_current: false, location:2, destination:8, has_activity_notification:False});home_nav.click({location:2})" aria-controls="" data-controller=" " data-s-popover-placement="right" aria-current="false" data-s-popover-auto-show="true" data-s-popover-hide-on-outside-click="never" > <div class="d-flex ai-center"> <svg aria-hidden="true" class="svg-icon iconHome" width="18" height="18" viewBox="0 0 18 18"><path d="M15 10v5a2 2 0 0 1-2 2H5a2 2 0 0 1-2-2v-5H0l9-9 9 9zm-8 1v6h4v-6z"/></svg> <span class="-link--channel-name pl6">Home</span> </div> </a> </li> <li class="ps-relative youarehere" aria-current="true"> <a id="nav-questions" href="/questions" class="s-block-link pl8 js-gps-track nav-links--link -link__with-icon" data-gps-track="top_nav.click({is_current: true, location:2, destination:1, has_activity_notification:False})" aria-controls="" data-controller=" " data-s-popover-placement="right" aria-current="false" data-s-popover-auto-show="true" data-s-popover-hide-on-outside-click="never" > <div class="d-flex ai-center"> <svg aria-hidden="true" class="svg-icon iconQuestion" width="18" height="18" viewBox="0 0 18 18"><path d="m4 15-3 3V4c0-1.1.9-2 2-2h12c1.09 0 2 .91 2 2v9c0 1.09-.91 2-2 2zm7.75-3.97c.72-.83.98-1.86.98-2.94 0-1.65-.7-3.22-2.3-3.83a4.4 4.4 0 0 0-3.02 0 3.8 3.8 0 0 0-2.32 3.83q0 1.93 1.03 3a3.8 3.8 0 0 0 2.85 1.07q.94 0 1.71-.34.97.66 1.06.7.34.2.7.3l.59-1.13a5 5 0 0 1-1.28-.66m-1.27-.9a5 5 0 0 0-1.5-.8l-.45.9q.5.18.98.5-.3.1-.65.11-.92 0-1.52-.68c-.86-1-.86-3.12 0-4.11.8-.9 2.35-.9 3.15 0 .9 1.01.86 3.03-.01 4.08"/></svg> <span class="-link--channel-name pl6">Questions</span> </div> </a> </li> <li class="ps-relative" aria-current="false"> <a href="/tags" class="s-block-link pl8 js-gps-track nav-links--link -link__with-icon" data-gps-track="top_nav.click({is_current: false, location:2, destination:2, has_activity_notification:False})" aria-controls="" data-controller=" " data-s-popover-placement="right" aria-current="false" data-s-popover-auto-show="true" data-s-popover-hide-on-outside-click="never" > <div class="d-flex ai-center"> <svg aria-hidden="true" class="svg-icon iconTags" width="18" height="18" viewBox="0 0 18 18"><path d="M9.24 1a3 3 0 0 0-2.12.88l-5.7 5.7a2 2 0 0 0-.38 2.31 3 3 0 0 1 .67-1.01l6-6A3 3 0 0 1 9.83 2H14a3 3 0 0 1 .79.1A2 2 0 0 0 13 1z" opacity=".4"/><path d="M9.83 3a2 2 0 0 0-1.42.59l-6 6a2 2 0 0 0 0 2.82L6.6 16.6a2 2 0 0 0 2.82 0l6-6A2 2 0 0 0 16 9.17V5a2 2 0 0 0-2-2zM12 9a2 2 0 1 1 0-4 2 2 0 0 1 0 4"/></svg> <span class="-link--channel-name pl6">Tags</span> </div> </a> </li> <li class="pb24"></li> <li class="ps-relative" aria-current="false"> <a id="nav-users" href="/users" class="s-block-link pl8 js-gps-track nav-links--link -link__with-icon" data-gps-track="top_nav.click({is_current: false, location:2, destination:3, has_activity_notification:False})" aria-controls="" data-controller=" " data-s-popover-placement="right" aria-current="false" data-s-popover-auto-show="true" data-s-popover-hide-on-outside-click="never" > <div class="d-flex ai-center"> <svg aria-hidden="true" class="svg-icon iconPeople" width="18" height="18" viewBox="0 0 18 18"><path d="M17 14c0 .44-.45 1-1 1H9a1 1 0 0 1-1-1H2c-.54 0-1-.56-1-1 0-2.63 3-4 3-4s.23-.4 0-1c-.84-.62-1.06-.59-1-3s1.37-3 2.5-3 2.44.58 2.5 3-.16 2.38-1 3c-.23.59 0 1 0 1s1.55.71 2.42 2.09c.78-.72 1.58-1.1 1.58-1.1s.23-.4 0-1c-.84-.61-1.06-.58-1-3s1.37-3 2.5-3 2.44.59 2.5 3c.05 2.42-.16 2.39-1 3-.23.6 0 1 0 1s3 1.38 3 4"/></svg> <span class="-link--channel-name pl6">Users</span> </div> </a> </li> <li class="ps-relative" aria-current="false"> <a id="nav-unanswered" href="/unanswered" class="s-block-link pl8 js-gps-track nav-links--link -link__with-icon" data-gps-track="top_nav.click({is_current: false, location:2, destination:5, has_activity_notification:False})" aria-controls="" data-controller=" " data-s-popover-placement="right" aria-current="false" data-s-popover-auto-show="true" data-s-popover-hide-on-outside-click="never" > <div class="d-flex ai-center"> <svg aria-hidden="true" class="svg-icon iconAnswer" width="18" height="18" viewBox="0 0 18 18"><path d="M14 15H3c-1.09 0-2-.91-2-2V4c0-1.1.9-2 2-2h12c1.09 0 2 .91 2 2v14zm-1.02-3L9.82 4H8.14l-3.06 8h1.68l.65-1.79h3.15l.69 1.79zm-2.93-3.12H7.9l1.06-2.92z"/></svg> <span class="-link--channel-name pl6">Unanswered</span> </div> </a> </li> </ol> </li> <li class="js-freemium-cta ps-relative mt32 mb8"> <div class="fs-fine tt-uppercase fc-black-600 fw-bold ml8 mt16 mb8">Teams</div> <div class="px12 pt12 pb4 mb12 fc-medium overflow-hidden"> <img class="wmx100 mx-auto mb12 h-auto d-block" width="151" height="24" src="https://cdn.sstatic.net/Img/teams/teams-promo.svg?v=e507948b81bf" alt=""> <p class="fs-fine"> Ask questions, find answers and collaborate at work with Stack Overflow for Teams. </p> <a href="https://stackoverflowteams.com/teams/create/free/?utm_medium=referral&utm_source=math-community&utm_campaign=side-bar&utm_content=explore-teams" class="w100 s-btn s-btn__filled s-btn__xs bg-orange-400 h:bg-orange-500 js-gps-track pt8 pr7 pb6 pl7" data-gps-track="teams.create.left-sidenav.click({ Action: 6 })" data-ga="["teams left navigation - anonymous","left nav free cta","stackoverflow.com/teams/create/free",null,null]">Try Teams for free</a> <a href="https://stackoverflow.co/teams/?utm_medium=referral&utm_source=math-community&utm_campaign=side-bar&utm_content=explore-teams" class="w100 s-btn s-btn__muted s-btn__xs mt1 js-gps-track" data-gps-track="teams.create.left-sidenav.click({ Action: 5 })" data-ga="["teams left navigation - anonymous","left nav free cta","stackoverflow.com/teams",null,null]">Explore Teams</a> </div> </li> <li class="d-flex ai-center jc-space-between ml8 mt32 mb8 js-create-team-cta d-none"> <a href="javascript:void(0)" class="s-link d-flex fl-grow1 fc-black-400 h:fc-black-600 fs-fine js-gps-track" role="button" aria-controls="popover-teams-create-cta" data-controller="s-popover" data-action="s-popover#toggle" data-s-popover-placement="bottom-start" data-s-popover-toggle-class="is-selected" data-gps-track="teams.create.left-sidenav.click({ Action: ShowInfo })" data-ga="["teams left navigation - anonymous","left nav show teams info",null,null,null]" > <div class="flex--item fl-grow1 fc-black-600 fw-bold tt-uppercase">Teams</div> <div class="flex--item px12"> <svg aria-hidden="true" class="svg-icon iconPlusSm" width="14" height="14" viewBox="0 0 14 14"><path d="M8 2H6v4H2v2h4v4h2V8h4V6H8z"/></svg> </div> </a> </li> <li class="ps-relative js-create-team-cta d-none"> <p class="fs-fine pr8 pl8 pb4 fc-black-400"> Ask questions, find answers and collaborate at work with Stack Overflow for Teams. <a href="https://stackoverflow.co/teams/?utm_medium=referral&utm_source=math-community&utm_campaign=side-bar&utm_content=explore-teams-compact" class="s-link s-link__grayscale s-link__underlined fw-bold">Explore Teams</a> </p> </li> </ol> </nav> </div> <div class="s-popover ws2" id="popover-teams-create-cta" role="menu" aria-hidden="true"> <div class="s-popover--arrow"></div> <div class="ps-relative overflow-hidden"> <p class="mb2"><strong>Teams</strong></p> <p class="mb12 fs-caption fc-black-400">Q&A for work</p> <p class="mb12 fs-caption fc-black-500">Connect and share knowledge within a single location that is structured and easy to search.</p> <a href="https://stackoverflow.co/teams/" class="js-gps-track s-btn s-btn__filled s-btn__xs" data-gps-track="teams.create.left-sidenav.click({ Action: CtaClick })" data-ga="["teams left navigation - anonymous","left nav cta","stackoverflow.com/teams",null,null]"> Learn more about Teams </a> </div> <div class="ps-absolute t8 r8"> <svg aria-hidden="true" class="fc-orange-400 svg-spot spotPeople" width="48" height="48" viewBox="0 0 48 48"><path d="M13.5 28a4.5 4.5 0 1 0 0-9 4.5 4.5 0 0 0 0 9M7 30a1 1 0 0 1 1-1h11a1 1 0 0 1 1 1v5h11v-5a1 1 0 0 1 1-1h12a1 1 0 0 1 1 1v10a2 2 0 0 1-2 2H33v5a1 1 0 0 1-1 1H20a1 1 0 0 1-1-1v-5H8a1 1 0 0 1-1-1zm25-6.5a4.5 4.5 0 1 0 9 0 4.5 4.5 0 0 0-9 0M24.5 34a4.5 4.5 0 1 0 0-9 4.5 4.5 0 0 0 0 9" opacity=".2"/><path d="M16.4 26.08A6 6 0 1 0 7.53 26C5.64 26.06 4 27.52 4 29.45V40a1 1 0 0 0 1 1h9a1 1 0 1 0 0-2h-4v-7a1 1 0 1 0-2 0v7H6v-9.55c0-.73.67-1.45 1.64-1.45H16a1 1 0 0 0 .4-1.92M12 18a4 4 0 1 1 0 8 4 4 0 0 1 0-8m16.47 14a6 6 0 1 0-8.94 0A3.6 3.6 0 0 0 16 35.5V46a1 1 0 0 0 1 1h14a1 1 0 0 0 1-1V35.5c0-1.94-1.64-3.42-3.53-3.5M20 28a4 4 0 1 1 8 0 4 4 0 0 1-8 0m-.3 6h8.6c1 0 1.7.75 1.7 1.5V45h-2v-7a1 1 0 1 0-2 0v7h-4v-7a1 1 0 1 0-2 0v7h-2v-9.5c0-.75.7-1.5 1.7-1.5M42 22c0 1.54-.58 2.94-1.53 4A3.5 3.5 0 0 1 44 29.45V40a1 1 0 0 1-1 1h-9a1 1 0 1 1 0-2h4v-7a1 1 0 1 1 2 0v7h2v-9.55A1.5 1.5 0 0 0 40.48 28H32a1 1 0 0 1-.4-1.92A6 6 0 1 1 42 22m-2 0a4 4 0 1 0-8 0 4 4 0 0 0 8 0"/><g opacity=".35"><path d="M17 10a1 1 0 011-1h12a1 1 0 110 2H18a1 1 0 01-1-1m1-5a1 1 0 100 2h12a1 1 0 100-2zM14 1a1 1 0 00-1 1v12a1 1 0 001 1h5.09l4.2 4.2a1 1 0 001.46-.04l3.7-4.16H34a1 1 0 001-1V2a1 1 0 00-1-1zm1 12V3h18v10h-5a1 1 0 00-.75.34l-3.3 3.7-3.74-3.75a1 1 0 00-.71-.29z"/></g></svg> </div> </div> </div> <div id="content" class=""> <div itemprop="mainEntity" itemscope itemtype="https://schema.org/Question"> <link itemprop="image" href="https://cdn.sstatic.net/Sites/math/Img/apple-touch-icon.png?v=0ae50baa40ed"> <div class="inner-content clearfix"> <div id="question-header" class="d-flex sm:fd-column"> <h1 itemprop="name" class="fs-headline1 ow-break-word mb8 flex--item fl1"><a href="/questions/1290529/difference-between-boundary-point-limit-point" class="question-hyperlink">Difference between boundary point & limit point.</a></h1> <div class="ml12 aside-cta flex--item sm:ml0 sm:mb12 sm:order-first d-flex jc-end"> <div class="ml12 aside-cta flex--item print:d-none"> <a href="/questions/ask" class="ws-nowrap s-btn s-btn__filled"> Ask Question </a> </div> </div> </div> <div class="d-flex fw-wrap pb8 mb16 bb bc-black-200"> <div class="flex--item ws-nowrap mr16 mb8" title="2015-05-20 02:51:28Z"> <span class="fc-black-400 mr2">Asked</span> <time itemprop="dateCreated" datetime="2015-05-20T02:51:28">9 years, 9 months ago</time> </div> <div class="flex--item ws-nowrap mr16 mb8"> <span class="fc-black-400 mr2">Modified</span> <a href="?lastactivity" class="s-link s-link__inherit" title="2019-05-08 14:17:22Z">5 years, 9 months ago</a> </div> <div class="flex--item ws-nowrap mb8" title="Viewed 23,727 times"> <span class="fc-black-400 mr2">Viewed</span> 24k times </div> </div> <div id="mainbar" role="main" aria-label="question and answers"> <div class="question js-question" data-questionid="1290529" data-position-on-page="0" data-score="14" id="question"> <style> </style> <div class="js-zone-container zone-container-main"> <div id="dfp-tlb" class="everyonelovesstackoverflow everyoneloves__top-leaderboard everyoneloves__leaderboard"></div> <div class="js-report-ad-button-container " style="width: 728px"></div> </div> <div class="post-layout "> <div class="votecell post-layout--left"> <div class="js-voting-container d-flex jc-center fd-column ai-center gs4 fc-black-300" data-post-id="1290529" data-referrer="None"> <button class="js-vote-up-btn flex--item s-btn s-btn__muted s-btn__outlined bar-pill bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200" id="upvote-btn-1290529" data-controller="s-tooltip" data-s-tooltip-placement="right" title="This question shows research effort; it is useful and clear" aria-pressed="false" aria-label="Up vote" data-selected-classes="fc-theme-primary-100 bc-theme-primary-500 bg-theme-primary-500" data-unselected-classes="bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200"> <svg aria-hidden="true" class="svg-icon iconArrowUp" width="18" height="18" viewBox="0 0 18 18"><path d="M1 12h16L9 4z"/></svg> </button> <input type="hidden" id="voteUpHash" value="69:3:31e,16:680d2ea3255e6c6e,10:1740539661,16:b075cdee3c44bbaa,7:1290529,14308d081704df175f027472bafd8ff181721feb6bfbc0048e00fc2c7b64c53c" /> <div class="js-vote-count flex--item d-flex fd-column ai-center fc-theme-body-font fw-bold fs-subheading py4" itemprop="upvoteCount" data-value="14"> 14 </div> <button class="js-vote-down-btn js-vote-down-question flex--item mb8 s-btn s-btn__muted s-btn__outlined bar-pill bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200" title="This question does not show any research effort; it is unclear or not useful" aria-pressed="false" aria-label="Down vote" data-selected-classes="fc-theme-primary-100 bc-theme-primary-500 bg-theme-primary-500" data-unselected-classes="bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200"> <svg aria-hidden="true" class="svg-icon iconArrowDown" width="18" height="18" viewBox="0 0 18 18"><path d="M1 6h16l-8 8z"/></svg> </button> <input type="hidden" id="voteDownHash" value="69:3:31e,16:d230b72b8f6c7b55,10:1740539661,16:844509d37b2c517c,7:1290529,3308580970a6d254d6628fc876f440443aab8c676b8a6b5be528b1a4594b6963" /> <button class="js-saves-btn s-btn s-btn__unset c-pointer py4" type="button" id="saves-btn-1290529" data-controller="s-tooltip" data-s-tooltip-placement="right" data-s-popover-placement="" title="Save this question." data-is-saved="false" aria-label="Save" data-post-id="1290529" data-post-type-id="1" data-user-privilege-for-post-click="0" aria-controls="" data-s-popover-auto-show="false" > <svg aria-hidden="true" class="fc-theme-primary-400 js-saves-btn-selected d-none svg-icon iconBookmark" width="18" height="18" viewBox="0 0 18 18"><path d="M3 17V3c0-1.1.9-2 2-2h8a2 2 0 0 1 2 2v14l-6-4z"/></svg> <svg aria-hidden="true" class="js-saves-btn-unselected svg-icon iconBookmarkAlt" width="18" height="18" viewBox="0 0 18 18"><path d="m9 10.6 4 2.66V3H5v10.26zM3 17V3c0-1.1.9-2 2-2h8a2 2 0 0 1 2 2v14l-6-4z"/></svg> </button> <a class="js-post-issue flex--item s-btn s-btn__unset c-pointer py6 mx-auto" href="/posts/1290529/timeline" data-shortcut="T" data-ks-title="timeline" data-controller="s-tooltip" data-s-tooltip-placement="right" title="Show activity on this post." aria-label="Timeline"><svg aria-hidden="true" class="mln2 mr0 svg-icon iconHistory" width="19" height="18" viewBox="0 0 19 18"><path d="M3 9a8 8 0 1 1 3.73 6.77L8.2 14.3A6 6 0 1 0 5 9l3.01-.01-4 4-4-4zm7-4h1.01L11 9.36l3.22 2.1-.6.93L10 10z"/></svg></a> </div> </div> <div class="postcell post-layout--right"> <span class="d-none">$\begingroup$</span> <div class="s-prose js-post-body" itemprop="text"> <p>A limit point is just a accumulation point whose neighbourhood contains infinitely many elements of the sequence. </p> <p>Is there any difference between boundary point & limit point? I've read in another question here that all boudary points are limit points, but is the converse true?</p> </div> <div class="mt24 mb12"> <div class="post-taglist d-flex gs4 gsy fd-column"> <div class="d-flex ps-relative fw-wrap"> <ul class='ml0 list-ls-none js-post-tag-list-wrapper d-inline'><li class='d-inline mr4 js-post-tag-list-item'><a href="/questions/tagged/real-analysis" class="s-tag post-tag" title="show questions tagged 'real-analysis'" aria-label="show questions tagged 'real-analysis'" rel="tag" aria-labelledby="tag-real-analysis-tooltip-container" data-tag-menu-origin="Unknown">real-analysis</a></li></ul> </div> </div> </div> <div class="mb0 "> <div class="mt16 d-flex gs8 gsy fw-wrap jc-end ai-start pt4 mb16"> <div class="flex--item mr16 fl1 w96"> <div class="js-post-menu pt2" data-post-id="1290529" data-post-type-id="1"> <div class="d-flex gs8 s-anchors s-anchors__muted fw-wrap"> <div class="flex--item"> <a href="/q/1290529" rel="nofollow" itemprop="url" class="js-share-link js-gps-track" title="Short permalink to this question" data-gps-track="post.click({ item: 2, priv: 0, post_type: 1 })" data-controller="se-share-sheet" data-se-share-sheet-title="Share a link to this question" data-se-share-sheet-subtitle="" data-se-share-sheet-post-type="question" data-se-share-sheet-social="facebook twitter " data-se-share-sheet-location="1" data-se-share-sheet-license-url="https%3a%2f%2fcreativecommons.org%2flicenses%2fby-sa%2f3.0%2f" data-se-share-sheet-license-name="CC BY-SA 3.0" data-s-popover-placement="bottom-start">Share</a> </div> <div class="flex--item"> <button type="button" class="js-cite-link s-btn s-btn__link">Cite</button> </div> <div class="flex--item"> <button type="button" id="btnFollowPost-1290529" class="s-btn s-btn__link js-follow-post js-follow-question js-gps-track" data-gps-track="post.click({ item: 14, priv: 0, post_type: 1 })" data-controller="s-tooltip " data-s-tooltip-placement="bottom" data-s-popover-placement="bottom" aria-controls="" title="Follow this question to receive notifications"> Follow <input type="hidden" id="voteFollowHash" value="69:3:31e,16:ac19d1418dcd8c70,10:1740539661,16:3f5649af33fbaf55,7:1290529,68803bbbd8d7c35fcb994a4b461ff3878a7fcdc38909d793546bfe5c30512d77" /> </button> </div> </div> <div class="js-menu-popup-container"></div> </div> </div> <div class="post-signature owner flex--item"> <div class="user-info "> <div class="d-flex "> <div class="user-action-time fl-grow1"> asked <span title='2015-05-20 02:51:28Z' class='relativetime'>May 20, 2015 at 2:51</span> </div> </div> <div class="user-gravatar32"> <span class="anonymous-gravatar"></span> </div> <div class="user-details" itemprop="author" itemscope itemtype="http://schema.org/Person"> user142971<span class="d-none" itemprop="name">user142971</span> <div class="-flair"> </div> </div> </div> </div> </div> </div> <span class="d-none">$\endgroup$</span> </div> <span class="d-none" itemprop="commentCount">5</span> <div class="post-layout--right js-post-comments-component"> <div id="comments-1290529" class="comments js-comments-container bt bc-black-200 mt12 " data-post-id="1290529" data-min-length="15"> <ul class="comments-list js-comments-list" data-remaining-comments-count="0" data-canpost="false" data-cansee="true" data-comments-unavailable="false" data-addlink-disabled="true"> <li id="comment-2620269" class="comment js-comment " data-comment-id="2620269" data-comment-owner-id="-1" data-comment-score="7"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-score js-comment-edit-hide"> <span title="number of 'useful comment' votes received" class="warm">7</span> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">No. Limit points can also be interior points.</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <span class="comment-user">user137731</span> </div> <span class="comment-date" dir="ltr"> <span class="v-visible-sr">Commented</span> <span title='2015-05-20 02:55:37Z, License: CC BY-SA 3.0' class='relativetime-clean'>May 20, 2015 at 2:55</span> </span> </div> </div> </li> <li id="comment-2620276" class="comment js-comment " data-comment-id="2620276" data-comment-owner-id="-1" data-comment-score="1"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-score js-comment-edit-hide"> <span title="number of 'useful comment' votes received" class="cool">1</span> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">@Bye_World: That answers my question:)</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <span class="comment-user">user142971</span> </div> <span class="comment-date" dir="ltr"> <span class="v-visible-sr">Commented</span> <span title='2015-05-20 02:57:51Z, License: CC BY-SA 3.0' class='relativetime-clean'>May 20, 2015 at 2:57</span> </span> </div> </div> </li> <li id="comment-2620277" class="comment js-comment " data-comment-id="2620277" data-comment-owner-id="-1" data-comment-score="0"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-score js-comment-edit-hide"> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">@Bye_World: Just can you tell me why is boundary point a limit point? Does its neighbourhood contains infinitely many elements of a sequence?</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <span class="comment-user">user142971</span> </div> <span class="comment-date" dir="ltr"> <span class="v-visible-sr">Commented</span> <span title='2015-05-20 03:00:05Z, License: CC BY-SA 3.0' class='relativetime-clean'>May 20, 2015 at 3:00</span> </span> </div> </div> </li> <li id="comment-2620284" class="comment js-comment " data-comment-id="2620284" data-comment-owner-id="-1" data-comment-score="0"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-score js-comment-edit-hide"> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">Give me a minute to type up an answer.</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <span class="comment-user">user137731</span> </div> <span class="comment-date" dir="ltr"> <span class="v-visible-sr">Commented</span> <span title='2015-05-20 03:04:45Z, License: CC BY-SA 3.0' class='relativetime-clean'>May 20, 2015 at 3:04</span> </span> </div> </div> </li> <li id="comment-6093072" class="comment js-comment " data-comment-id="6093072" data-comment-owner-id="157069" data-comment-score="0"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-score js-comment-edit-hide"> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">According to Wikipedia, a limit point and an accumulation point are synonymous. A boundary point is different.</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <a href="/users/157069/clay" title="2,853 reputation" class="comment-user">clay</a> </div> <span class="comment-date" dir="ltr"> <span class="v-visible-sr">Commented</span> <span title='2018-10-10 22:59:20Z, License: CC BY-SA 4.0' class='relativetime-clean'>Oct 10, 2018 at 22:59</span> </span> </div> </div> </li> </ul> </div> <div id="comments-link-1290529" data-rep=50 data-anon=true> <a class="js-add-link comments-link disabled-link" title="Use comments to ask for more information or suggest improvements. Avoid answering questions in comments." href="#" role="button">Add a comment</a> <span class="js-link-separator dno"> | </span> <a class="js-show-link comments-link dno" title="Expand to show all comments on this post" href=# onclick="" role="button"></a> </div> </div> <script defer src="https://cdn.sstatic.net/Js/webpack-chunks/svelte.en.js?v=f1f216c36f5e"></script><script defer src="https://cdn.sstatic.net/Js/webpack-chunks/stacks-svelte.en.js?v=e5a0c3f4bb1b"></script><script defer src="https://cdn.sstatic.net/Js/webpack-chunks/2784.en.js?v=414ac3934ec2"></script><script defer src="https://cdn.sstatic.net/Js/islands/comment-popover.en.js?v=3f9e19615dd5"></script> </div> </div> <div id="answers"> <a name="tab-top"></a> <div id="answers-header"> <div class="answers-subheader d-flex ai-center mb8"> <div class="flex--item fl1"> <h2 class="mb0" data-answercount="3"> 3 Answers <span style="display:none;" itemprop="answerCount">3</span> </h2> </div> <div class="flex--item"> <div class="d-flex g4 gsx ai-center sm:fd-column sm:ai-start"> <div class="d-flex fd-column ai-end sm:ai-start"> <label class="flex--item fs-caption" for="answer-sort-dropdown-select-menu"> Sorted by: </label> <a class="js-sort-preference-change s-link flex--item fs-fine d-none" data-value="ScoreDesc" href="/questions/1290529/difference-between-boundary-point-limit-point?answertab=scoredesc#tab-top" > Reset to default </a> </div> <div class="flex--item s-select"> <select id="answer-sort-dropdown-select-menu"> <option value=scoredesc selected=selected > Highest score (default) </option> <option value=modifieddesc > Date modified (newest first) </option> <option value=createdasc > Date created (oldest first) </option> </select> </div> </div> </div> </div> </div> <a name="1290541"></a> <div id="answer-1290541" class="answer js-answer accepted-answer js-accepted-answer" data-answerid="1290541" data-parentid="1290529" data-score="21" data-position-on-page="1" data-highest-scored="1" data-question-has-accepted-highest-score="1" itemprop="acceptedAnswer" itemscope itemtype="https://schema.org/Answer"> <div class="post-layout"> <div class="votecell post-layout--left"> <div class="js-voting-container d-flex jc-center fd-column ai-center gs4 fc-black-300" data-post-id="1290541" data-referrer="None"> <button class="js-vote-up-btn flex--item s-btn s-btn__muted s-btn__outlined bar-pill bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200" id="upvote-btn-1290541" data-controller="s-tooltip" data-s-tooltip-placement="right" title="This answer is useful" aria-pressed="false" aria-label="Up vote" data-selected-classes="fc-theme-primary-100 bc-theme-primary-500 bg-theme-primary-500" data-unselected-classes="bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200"> <svg aria-hidden="true" class="svg-icon iconArrowUp" width="18" height="18" viewBox="0 0 18 18"><path d="M1 12h16L9 4z"/></svg> </button> <input type="hidden" id="voteUpHash" value="69:3:31e,16:f7283ac34ca2b812,10:1740539661,16:c70c69a241869ce7,7:1290541,7f3106dbdb806f883f887dfee9e2091560acfab825c70045ebeb1f5e0fb3a89a" /> <div class="js-vote-count flex--item d-flex fd-column ai-center fc-theme-body-font fw-bold fs-subheading py4" itemprop="upvoteCount" data-value="21"> 21 </div> <button class="js-vote-down-btn flex--item mb8 s-btn s-btn__muted s-btn__outlined bar-pill bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200" title="This answer is not useful" aria-pressed="false" aria-label="Down vote" data-selected-classes="fc-theme-primary-100 bc-theme-primary-500 bg-theme-primary-500" data-unselected-classes="bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200"> <svg aria-hidden="true" class="svg-icon iconArrowDown" width="18" height="18" viewBox="0 0 18 18"><path d="M1 6h16l-8 8z"/></svg> </button> <input type="hidden" id="voteDownHash" value="69:3:31e,16:068e31bfd85c1643,10:1740539661,16:e967292a67f1bc3e,7:1290541,65bc557dfda1e42860d8ecb92daeb7612cec69df1011c6c6e9827de2006e8e30" /> <button class="js-saves-btn s-btn s-btn__unset c-pointer py4" type="button" id="saves-btn-1290541" data-controller="s-tooltip" data-s-tooltip-placement="right" data-s-popover-placement="" title="Save this answer." data-is-saved="false" aria-label="Save" data-post-id="1290541" data-post-type-id="2" data-user-privilege-for-post-click="0" aria-controls="" data-s-popover-auto-show="false" > <svg aria-hidden="true" class="fc-theme-primary-400 js-saves-btn-selected d-none svg-icon iconBookmark" width="18" height="18" viewBox="0 0 18 18"><path d="M3 17V3c0-1.1.9-2 2-2h8a2 2 0 0 1 2 2v14l-6-4z"/></svg> <svg aria-hidden="true" class="js-saves-btn-unselected svg-icon iconBookmarkAlt" width="18" height="18" viewBox="0 0 18 18"><path d="m9 10.6 4 2.66V3H5v10.26zM3 17V3c0-1.1.9-2 2-2h8a2 2 0 0 1 2 2v14l-6-4z"/></svg> </button> <div class="js-accepted-answer-indicator flex--item fc-green-400 py6 mtn8" data-s-tooltip-placement="right" title="Loading when this answer was accepted…" tabindex="0" role="note" aria-label="Accepted"> <div class="ta-center"> <svg aria-hidden="true" class="svg-icon iconCheckmarkLg" width="36" height="36" viewBox="0 0 36 36"><path d="m6 14 8 8L30 6v8L14 30l-8-8z"/></svg> </div> </div> <a class="js-post-issue flex--item s-btn s-btn__unset c-pointer py6 mx-auto" href="/posts/1290541/timeline" data-shortcut="T" data-ks-title="timeline" data-controller="s-tooltip" data-s-tooltip-placement="right" title="Show activity on this post." aria-label="Timeline"><svg aria-hidden="true" class="mln2 mr0 svg-icon iconHistory" width="19" height="18" viewBox="0 0 19 18"><path d="M3 9a8 8 0 1 1 3.73 6.77L8.2 14.3A6 6 0 1 0 5 9l3.01-.01-4 4-4-4zm7-4h1.01L11 9.36l3.22 2.1-.6.93L10 10z"/></svg></a> </div> </div> <div class="answercell post-layout--right"> <span class="d-none">$\begingroup$</span> <div class="s-prose js-post-body" itemprop="text"> <p>Definition of <em>Limit Point</em>: "Let $S$ be a subset of a topological space $X$. A point $x$ in $X$ is a limit point of $S$ if every neighbourhood of $x$ contains at least one point of $S$ different from $x$ itself."<br><em>~from <a href="http://en.wikipedia.org/wiki/Limit_point" rel="noreferrer">Wikipedia</a></em></p> <p>Definition of <em>Boundary</em>: "Let $S$ be a subset of a topological space $X$. The boundary of $S$ is the set of points $p$ of $X$ such that every neighborhood of $p$ contains at least one point of $S$ and at least one point not of $S$." <br><em>~from <a href="http://en.wikipedia.org/wiki/Boundary_%28topology%29" rel="noreferrer">Wikipedia</a></em></p> <p>So <strong>deleted</strong> neighborhoods of limit points must contain at least one point <em>in</em> $S$. But (not necessarily deleted) neighborhoods of boundary points must contain at least one point <em>in</em> $S$ <strong>AND</strong> one point <em>not in</em> $S$.</p> <p>So they are not the same.</p> <p>Consider the set $S=\{0\}$ in $\Bbb R$ with the usual topology. $0$ is a boundary point but NOT a limit point of $S$.</p> <p>Consider the set $S'=[0,1]$ in $\Bbb R$ with the usual topology. $0.5$ is a limit point but NOT a boundary point of $S'$.</p> </div> <div class="mt24"> <div class="d-flex fw-wrap ai-start jc-end gs8 gsy"> <time itemprop="dateCreated" datetime="2015-05-20T03:08:37"></time> <div class="flex--item mr16" style="flex: 1 1 100px;"> <div class="js-post-menu pt2" data-post-id="1290541" data-post-type-id="2"> <div class="d-flex gs8 s-anchors s-anchors__muted fw-wrap"> <div class="flex--item"> <a href="/a/1290541" rel="nofollow" itemprop="url" class="js-share-link js-gps-track" title="Short permalink to this answer" data-gps-track="post.click({ item: 2, priv: 0, post_type: 2 })" data-controller="se-share-sheet" data-se-share-sheet-title="Share a link to this answer" data-se-share-sheet-subtitle="" data-se-share-sheet-post-type="answer" data-se-share-sheet-social="facebook twitter " data-se-share-sheet-location="2" data-se-share-sheet-license-url="https%3a%2f%2fcreativecommons.org%2flicenses%2fby-sa%2f3.0%2f" data-se-share-sheet-license-name="CC BY-SA 3.0" data-s-popover-placement="bottom-start">Share</a> </div> <div class="flex--item"> <button type="button" class="js-cite-link s-btn s-btn__link">Cite</button> </div> <div class="flex--item"> <button type="button" id="btnFollowPost-1290541" class="s-btn s-btn__link js-follow-post js-follow-answer js-gps-track" data-gps-track="post.click({ item: 14, priv: 0, post_type: 2 })" data-controller="s-tooltip " data-s-tooltip-placement="bottom" data-s-popover-placement="bottom" aria-controls="" title="Follow this answer to receive notifications"> Follow <input type="hidden" id="voteFollowHash" value="69:3:31e,16:0fb8d03d600d2d12,10:1740539661,16:cbb02309fd541cb1,7:1290541,f2f24cda6ae0dcb0c9ef415087bee92860438654a4cfd577359ad763754f8c93" /> </button> </div> </div> <div class="js-menu-popup-container"></div> </div> </div> <div class="post-signature flex--item fl0"> <div class="user-info "> <div class="d-flex "> <div class="user-action-time fl-grow1"> <a href="/posts/1290541/revisions" title="show all edits to this post" class="js-gps-track" data-gps-track="post.click({ item: 4, priv: 0, post_type: 2 })">edited <span title='2016-09-27 14:20:57Z' class='relativetime'>Sep 27, 2016 at 14:20</span></a> </div> </div> <div class="user-gravatar32"> </div> <div class="user-details"> <div class="-flair"> </div> </div> </div> </div> <div class="post-signature flex--item fl0"> <div class="user-info "> <div class="d-flex "> <div class="user-action-time fl-grow1"> answered <span title='2015-05-20 03:08:37Z' class='relativetime'>May 20, 2015 at 3:08</span> </div> </div> <div class="user-gravatar32"> <span class="anonymous-gravatar"></span> </div> <div class="user-details" itemprop="author" itemscope itemtype="http://schema.org/Person"> user137731<span class="d-none" itemprop="name">user137731</span> <div class="-flair"> </div> </div> </div> </div> </div> </div> <span class="d-none">$\endgroup$</span> </div> <span class="d-none" itemprop="commentCount">10</span> <div class="post-layout--right js-post-comments-component"> <div id="comments-1290541" class="comments js-comments-container bt bc-black-200 mt12 " data-post-id="1290541" data-min-length="15"> <ul class="comments-list js-comments-list" data-remaining-comments-count="5" data-canpost="false" data-cansee="true" data-comments-unavailable="false" data-addlink-disabled="true"> <li id="comment-2621281" class="comment js-comment " data-comment-id="2621281" data-comment-owner-id="15501" data-comment-score="0"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-score js-comment-edit-hide"> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">Let it be known that <i>boundary points need not be members of</i> $X$. For example, the set $T=(0,1)$ has two boundary points: $0$ and $1$, even though they are not members of $T$.</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <a href="/users/15501/chharvey" title="2,662 reputation" class="comment-user">chharvey</a> </div> <span class="comment-date" dir="ltr"> <span class="v-visible-sr">Commented</span> <span title='2015-05-20 13:15:20Z, License: CC BY-SA 3.0' class='relativetime-clean'>May 20, 2015 at 13:15</span> </span> <span title="this comment was edited 2 times"> <svg aria-hidden="true" class="va-text-bottom o50 svg-icon iconPencilSm" width="14" height="14" viewBox="0 0 14 14"><path fill="#F1B600" d="m2 10.12 6.37-6.43 1.88 1.88L3.88 12H2z"/><path fill="#E87C87" d="m11.1 1.71 1.13 1.12c.2.2.2.51 0 .71L11.1 4.7 9.21 2.86l1.17-1.15c.2-.2.51-.2.71 0"/></svg> </span> </div> </div> </li> <li id="comment-2621333" class="comment js-comment " data-comment-id="2621333" data-comment-owner-id="15501" data-comment-score="0"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-score js-comment-edit-hide"> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">Yes, that's true. From a purely answer-the-OP's-question point of view, it helps to make a distinction, but from an educational point of view (for others who might later stumble upon your answer) it should be technically correct.</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <a href="/users/15501/chharvey" title="2,662 reputation" class="comment-user">chharvey</a> </div> <span class="comment-date" dir="ltr"> <span class="v-visible-sr">Commented</span> <span title='2015-05-20 13:39:26Z, License: CC BY-SA 3.0' class='relativetime-clean'>May 20, 2015 at 13:39</span> </span> </div> </div> </li> <li id="comment-2621419" class="comment js-comment " data-comment-id="2621419" data-comment-owner-id="15501" data-comment-score="0"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-score js-comment-edit-hide"> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">My apologies. I thought $X$ was the set in question, not the entire topological space.</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <a href="/users/15501/chharvey" title="2,662 reputation" class="comment-user">chharvey</a> </div> <span class="comment-date" dir="ltr"> <span class="v-visible-sr">Commented</span> <span title='2015-05-20 14:03:12Z, License: CC BY-SA 3.0' class='relativetime-clean'>May 20, 2015 at 14:03</span> </span> </div> </div> </li> <li id="comment-2621437" class="comment js-comment " data-comment-id="2621437" data-comment-owner-id="-1" data-comment-score="0"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-score js-comment-edit-hide"> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">@chharvey No worries. :)</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <span class="comment-user">user137731</span> </div> <span class="comment-date" dir="ltr"> <span class="v-visible-sr">Commented</span> <span title='2015-05-20 14:11:07Z, License: CC BY-SA 3.0' class='relativetime-clean'>May 20, 2015 at 14:11</span> </span> </div> </div> </li> <li id="comment-3728361" class="comment js-comment " data-comment-id="3728361" data-comment-owner-id="88587" data-comment-score="0"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-score js-comment-edit-hide"> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">@Bye_World. Can you explain why for the set $S={0}$ in $\Bbb R$ , $0$ is not the limit point?</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <a href="/users/88587/ckm" title="1,642 reputation" class="comment-user">CKM</a> </div> <span class="comment-date" dir="ltr"> <span class="v-visible-sr">Commented</span> <span title='2016-06-13 08:29:12Z, License: CC BY-SA 3.0' class='relativetime-clean'>Jun 13, 2016 at 8:29</span> </span> </div> </div> </li> </ul> </div> <div id="comments-link-1290541" data-rep=50 data-anon=true> <a class="js-add-link comments-link dno" title="Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”." href="#" role="button"></a> <span class="js-link-separator dno"> | </span> <a class="js-show-link comments-link " title="Expand to show all comments on this post" href=# onclick="" role="button">Show <b>5</b> more comments</a> </div> </div> </div> </div> <div class="js-zone-container zone-container-main"> <div id="dfp-mlb" class="everyonelovesstackoverflow everyoneloves__mid-leaderboard everyoneloves__leaderboard"></div> <div class="js-report-ad-button-container " style="width: 728px"></div> </div> <a name="1290532"></a> <div id="answer-1290532" class="answer js-answer" data-answerid="1290532" data-parentid="1290529" data-score="3" data-position-on-page="2" data-highest-scored="0" data-question-has-accepted-highest-score="1" itemprop="suggestedAnswer" itemscope itemtype="https://schema.org/Answer"> <div class="post-layout"> <div class="votecell post-layout--left"> <div class="js-voting-container d-flex jc-center fd-column ai-center gs4 fc-black-300" data-post-id="1290532" data-referrer="None"> <button class="js-vote-up-btn flex--item s-btn s-btn__muted s-btn__outlined bar-pill bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200" id="upvote-btn-1290532" data-controller="s-tooltip" data-s-tooltip-placement="right" title="This answer is useful" aria-pressed="false" aria-label="Up vote" data-selected-classes="fc-theme-primary-100 bc-theme-primary-500 bg-theme-primary-500" data-unselected-classes="bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200"> <svg aria-hidden="true" class="svg-icon iconArrowUp" width="18" height="18" viewBox="0 0 18 18"><path d="M1 12h16L9 4z"/></svg> </button> <input type="hidden" id="voteUpHash" value="69:3:31e,16:6076210aac5c30d3,10:1740539661,16:f6166367258275eb,7:1290532,b690067374a2024095705198e6be915df8960980ce4dcc1a1ca31d4fe2ce6ef2" /> <div class="js-vote-count flex--item d-flex fd-column ai-center fc-theme-body-font fw-bold fs-subheading py4" itemprop="upvoteCount" data-value="3"> 3 </div> <button class="js-vote-down-btn flex--item mb8 s-btn s-btn__muted s-btn__outlined bar-pill bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200" title="This answer is not useful" aria-pressed="false" aria-label="Down vote" data-selected-classes="fc-theme-primary-100 bc-theme-primary-500 bg-theme-primary-500" data-unselected-classes="bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200"> <svg aria-hidden="true" class="svg-icon iconArrowDown" width="18" height="18" viewBox="0 0 18 18"><path d="M1 6h16l-8 8z"/></svg> </button> <input type="hidden" id="voteDownHash" value="69:3:31e,16:2991abca96b101b8,10:1740539661,16:eb3fc21cc761ada5,7:1290532,3d35c664d7f269d125d2439831366bf8b35af10ed955b59180fc1d64307690ce" /> <button class="js-saves-btn s-btn s-btn__unset c-pointer py4" type="button" id="saves-btn-1290532" data-controller="s-tooltip" data-s-tooltip-placement="right" data-s-popover-placement="" title="Save this answer." data-is-saved="false" aria-label="Save" data-post-id="1290532" data-post-type-id="2" data-user-privilege-for-post-click="0" aria-controls="" data-s-popover-auto-show="false" > <svg aria-hidden="true" class="fc-theme-primary-400 js-saves-btn-selected d-none svg-icon iconBookmark" width="18" height="18" viewBox="0 0 18 18"><path d="M3 17V3c0-1.1.9-2 2-2h8a2 2 0 0 1 2 2v14l-6-4z"/></svg> <svg aria-hidden="true" class="js-saves-btn-unselected svg-icon iconBookmarkAlt" width="18" height="18" viewBox="0 0 18 18"><path d="m9 10.6 4 2.66V3H5v10.26zM3 17V3c0-1.1.9-2 2-2h8a2 2 0 0 1 2 2v14l-6-4z"/></svg> </button> <div class="js-accepted-answer-indicator flex--item fc-green-400 py6 mtn8 d-none" data-s-tooltip-placement="right" title="Loading when this answer was accepted…" tabindex="0" role="note" aria-label="Accepted"> <div class="ta-center"> <svg aria-hidden="true" class="svg-icon iconCheckmarkLg" width="36" height="36" viewBox="0 0 36 36"><path d="m6 14 8 8L30 6v8L14 30l-8-8z"/></svg> </div> </div> <a class="js-post-issue flex--item s-btn s-btn__unset c-pointer py6 mx-auto" href="/posts/1290532/timeline" data-shortcut="T" data-ks-title="timeline" data-controller="s-tooltip" data-s-tooltip-placement="right" title="Show activity on this post." aria-label="Timeline"><svg aria-hidden="true" class="mln2 mr0 svg-icon iconHistory" width="19" height="18" viewBox="0 0 19 18"><path d="M3 9a8 8 0 1 1 3.73 6.77L8.2 14.3A6 6 0 1 0 5 9l3.01-.01-4 4-4-4zm7-4h1.01L11 9.36l3.22 2.1-.6.93L10 10z"/></svg></a> </div> </div> <div class="answercell post-layout--right"> <span class="d-none">$\begingroup$</span> <div class="s-prose js-post-body" itemprop="text"> <p>Consider the interval $[0,1]$. Each element of it is a limit point, i.e. $\alpha$ is a limit of the sequence $n_1=\alpha, n_2=\alpha, \ldots$. Only $0,1$ are boundary points.</p> </div> <div class="mt24"> <div class="d-flex fw-wrap ai-start jc-end gs8 gsy"> <time itemprop="dateCreated" datetime="2015-05-20T02:55:46"></time> <div class="flex--item mr16" style="flex: 1 1 100px;"> <div class="js-post-menu pt2" data-post-id="1290532" data-post-type-id="2"> <div class="d-flex gs8 s-anchors s-anchors__muted fw-wrap"> <div class="flex--item"> <a href="/a/1290532" rel="nofollow" itemprop="url" class="js-share-link js-gps-track" title="Short permalink to this answer" data-gps-track="post.click({ item: 2, priv: 0, post_type: 2 })" data-controller="se-share-sheet" data-se-share-sheet-title="Share a link to this answer" data-se-share-sheet-subtitle="" data-se-share-sheet-post-type="answer" data-se-share-sheet-social="facebook twitter " data-se-share-sheet-location="2" data-se-share-sheet-license-url="https%3a%2f%2fcreativecommons.org%2flicenses%2fby-sa%2f3.0%2f" data-se-share-sheet-license-name="CC BY-SA 3.0" data-s-popover-placement="bottom-start">Share</a> </div> <div class="flex--item"> <button type="button" class="js-cite-link s-btn s-btn__link">Cite</button> </div> <div class="flex--item"> <button type="button" id="btnFollowPost-1290532" class="s-btn s-btn__link js-follow-post js-follow-answer js-gps-track" data-gps-track="post.click({ item: 14, priv: 0, post_type: 2 })" data-controller="s-tooltip " data-s-tooltip-placement="bottom" data-s-popover-placement="bottom" aria-controls="" title="Follow this answer to receive notifications"> Follow <input type="hidden" id="voteFollowHash" value="69:3:31e,16:194e8752edb93b01,10:1740539661,16:1947f326c89731d6,7:1290532,56513a678a94d6bde76f1a132b342e8418e64fd8679566bdd649375c36fa19b2" /> </button> </div> </div> <div class="js-menu-popup-container"></div> </div> </div> <div class="post-signature flex--item fl0"> <div class="user-info user-hover "> <div class="d-flex "> <div class="user-action-time fl-grow1"> answered <span title='2015-05-20 02:55:46Z' class='relativetime'>May 20, 2015 at 2:55</span> </div> </div> <div class="user-gravatar32"> <a href="/users/73324/vadim123"><div class="gravatar-wrapper-32"><img src="https://www.gravatar.com/avatar/8c3092bcb13c4ed655051ea8bfefe555?s=64&d=identicon&r=PG" alt="vadim123's user avatar" width="32" height="32" class="bar-sm"></div></a> </div> <div class="user-details" itemprop="author" itemscope itemtype="http://schema.org/Person"> <a href="/users/73324/vadim123" dir="auto">vadim123</a><span class="d-none" itemprop="name">vadim123</span> <div class="-flair"> <span class="reputation-score" title="reputation score 83,557" dir="ltr">83.6k</span><span title="9 gold badges" aria-hidden="true"><span class="badge1"></span><span class="badgecount">9</span></span><span class="v-visible-sr">9 gold badges</span><span title="119 silver badges" aria-hidden="true"><span class="badge2"></span><span class="badgecount">119</span></span><span class="v-visible-sr">119 silver badges</span><span title="223 bronze badges" aria-hidden="true"><span class="badge3"></span><span class="badgecount">223</span></span><span class="v-visible-sr">223 bronze badges</span> </div> </div> </div> </div> </div> </div> <span class="d-none">$\endgroup$</span> </div> <span class="d-none" itemprop="commentCount"></span> <div class="post-layout--right js-post-comments-component"> <div id="comments-1290532" class="comments js-comments-container bt bc-black-200 mt12 dno" data-post-id="1290532" data-min-length="15"> <ul class="comments-list js-comments-list" data-remaining-comments-count="0" data-canpost="false" data-cansee="true" data-comments-unavailable="false" data-addlink-disabled="true"> </ul> </div> <div id="comments-link-1290532" data-rep=50 data-anon=true> <a class="js-add-link comments-link disabled-link" title="Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”." href="#" role="button">Add a comment</a> <span class="js-link-separator dno"> | </span> <a class="js-show-link comments-link dno" title="Expand to show all comments on this post" href=# onclick="" role="button"></a> </div> </div> </div> </div> <a name="3146487"></a> <div id="answer-3146487" class="answer js-answer" data-answerid="3146487" data-parentid="1290529" data-score="1" data-position-on-page="3" data-highest-scored="0" data-question-has-accepted-highest-score="1" itemprop="suggestedAnswer" itemscope itemtype="https://schema.org/Answer"> <div class="post-layout"> <div class="votecell post-layout--left"> <div class="js-voting-container d-flex jc-center fd-column ai-center gs4 fc-black-300" data-post-id="3146487" data-referrer="None"> <button class="js-vote-up-btn flex--item s-btn s-btn__muted s-btn__outlined bar-pill bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200" id="upvote-btn-3146487" data-controller="s-tooltip" data-s-tooltip-placement="right" title="This answer is useful" aria-pressed="false" aria-label="Up vote" data-selected-classes="fc-theme-primary-100 bc-theme-primary-500 bg-theme-primary-500" data-unselected-classes="bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200"> <svg aria-hidden="true" class="svg-icon iconArrowUp" width="18" height="18" viewBox="0 0 18 18"><path d="M1 12h16L9 4z"/></svg> </button> <input type="hidden" id="voteUpHash" value="69:3:31e,16:b8b2d4da14843a80,10:1740539661,16:51a9ccbc3e11fb58,7:3146487,f3ecca23e77c46170ff35a883d10eaaf73afed145522c92360928983fa6972a1" /> <div class="js-vote-count flex--item d-flex fd-column ai-center fc-theme-body-font fw-bold fs-subheading py4" itemprop="upvoteCount" data-value="1"> 1 </div> <button class="js-vote-down-btn flex--item mb8 s-btn s-btn__muted s-btn__outlined bar-pill bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200" title="This answer is not useful" aria-pressed="false" aria-label="Down vote" data-selected-classes="fc-theme-primary-100 bc-theme-primary-500 bg-theme-primary-500" data-unselected-classes="bc-black-225 f:bc-theme-secondary-400 f:bg-theme-secondary-400 f:fc-black-050 h:bg-theme-primary-200"> <svg aria-hidden="true" class="svg-icon iconArrowDown" width="18" height="18" viewBox="0 0 18 18"><path d="M1 6h16l-8 8z"/></svg> </button> <input type="hidden" id="voteDownHash" value="69:3:31e,16:a26153fd8e8993c8,10:1740539661,16:380caecc60272041,7:3146487,44990ed925fff942a9f3169b3cf9d67c5a3041a868052377e100fd574713c36f" /> <button class="js-saves-btn s-btn s-btn__unset c-pointer py4" type="button" id="saves-btn-3146487" data-controller="s-tooltip" data-s-tooltip-placement="right" data-s-popover-placement="" title="Save this answer." data-is-saved="false" aria-label="Save" data-post-id="3146487" data-post-type-id="2" data-user-privilege-for-post-click="0" aria-controls="" data-s-popover-auto-show="false" > <svg aria-hidden="true" class="fc-theme-primary-400 js-saves-btn-selected d-none svg-icon iconBookmark" width="18" height="18" viewBox="0 0 18 18"><path d="M3 17V3c0-1.1.9-2 2-2h8a2 2 0 0 1 2 2v14l-6-4z"/></svg> <svg aria-hidden="true" class="js-saves-btn-unselected svg-icon iconBookmarkAlt" width="18" height="18" viewBox="0 0 18 18"><path d="m9 10.6 4 2.66V3H5v10.26zM3 17V3c0-1.1.9-2 2-2h8a2 2 0 0 1 2 2v14l-6-4z"/></svg> </button> <div class="js-accepted-answer-indicator flex--item fc-green-400 py6 mtn8 d-none" data-s-tooltip-placement="right" title="Loading when this answer was accepted…" tabindex="0" role="note" aria-label="Accepted"> <div class="ta-center"> <svg aria-hidden="true" class="svg-icon iconCheckmarkLg" width="36" height="36" viewBox="0 0 36 36"><path d="m6 14 8 8L30 6v8L14 30l-8-8z"/></svg> </div> </div> <a class="js-post-issue flex--item s-btn s-btn__unset c-pointer py6 mx-auto" href="/posts/3146487/timeline" data-shortcut="T" data-ks-title="timeline" data-controller="s-tooltip" data-s-tooltip-placement="right" title="Show activity on this post." aria-label="Timeline"><svg aria-hidden="true" class="mln2 mr0 svg-icon iconHistory" width="19" height="18" viewBox="0 0 19 18"><path d="M3 9a8 8 0 1 1 3.73 6.77L8.2 14.3A6 6 0 1 0 5 9l3.01-.01-4 4-4-4zm7-4h1.01L11 9.36l3.22 2.1-.6.93L10 10z"/></svg></a> </div> </div> <div class="answercell post-layout--right"> <span class="d-none">$\begingroup$</span> <div class="s-prose js-post-body" itemprop="text"> <p>Well, as someone has figured it out by supplying the definitions of limit point and boundary point. Now if we just head toward the general set topological approach we will find that , if <span class="math-container">$\Bbb{S}$</span> <span class="math-container">${\subset}$</span> of <span class="math-container">$\Bbb{R}$</span> , and if <span class="math-container">$\Bbb{X}$</span> be the boundary then <span class="math-container">$\Bbb{X}$</span>=cl(S)~int ( S) . So if p is a boundary point, then p will be in <span class="math-container">$\Bbb{X}$</span> . And we call <span class="math-container">$\Bbb{S}$</span> a closed set if it contains all it's boundary points. Now as we also know it's equivalent definition that s will be a closed set if it contains all it limit point.</p> <p>But that doesn't not imply that a limit point is a boundary point as a limit point can also be a interior point . Let's check the proof.</p> <p>Let <span class="math-container">$\Bbb{S}$</span> is our set of which l is a int point . Then for `<span class="math-container">$\epsilon$</span>>0 , N(l, <span class="math-container">$\epsilon$</span> ) contained in l . Now we will try to prove it contrapositively . Let l is not an int point . Then N(l, <span class="math-container">$\epsilon$</span> ) is not contained in <span class="math-container">$\Bbb{S}$</span> . Now let, €>0 then either €< <span class="math-container">$\epsilon$</span> or €≥ <span class="math-container">$\epsilon$</span></p> <p>When, €< <span class="math-container">$\epsilon$</span> as N(l, <span class="math-container">$\epsilon$</span> ) is not contained in <span class="math-container">$\Bbb{S}$</span> , so N(l, €) is not also contained in s . It suggests that, N'(l,€) <span class="math-container">${\cap}$</span> <span class="math-container">$\Bbb{S}$</span> = <span class="math-container">$\phi$</span> So, l is not a limit point of <span class="math-container">$\Bbb{S}$</span></p> <p>When, €≥ <span class="math-container">$\epsilon$</span> , N(l, <span class="math-container">$\epsilon$</span> ) is contained in N(l, €). So from here also it can be shown that , <span class="math-container">$\Bbb{S}$</span> <span class="math-container">${\cap}$</span> N'(l,€) is <span class="math-container">$\phi$</span> . So l is not a limit point of <span class="math-container">$\Bbb{S}$</span> .</p> <p>So if l is not an int point of <span class="math-container">$\Bbb{S}$</span> , it's not an limit point of <span class="math-container">$\Bbb{S}$</span> . It implies that if l is an limit point of <span class="math-container">$\Bbb{S}$</span> , it's an interior point of <span class="math-container">$\Bbb{S}$</span> .</p> <p>Now, there are also some cases where the above assertion fails. So l may or may not belongs to cl ( <span class="math-container">$\Bbb{S}$</span> ) ~ int ( <span class="math-container">$\Bbb{S}$</span> ) </p> <p>And the whole discussion tells us that a limit point can be a boundary point but that doesn't mean every limit point is a boundary point. And that's it !!!</p> </div> <div class="mt24"> <div class="d-flex fw-wrap ai-start jc-end gs8 gsy"> <time itemprop="dateCreated" datetime="2019-03-13T11:51:52"></time> <div class="flex--item mr16" style="flex: 1 1 100px;"> <div class="js-post-menu pt2" data-post-id="3146487" data-post-type-id="2"> <div class="d-flex gs8 s-anchors s-anchors__muted fw-wrap"> <div class="flex--item"> <a href="/a/3146487" rel="nofollow" itemprop="url" class="js-share-link js-gps-track" title="Short permalink to this answer" data-gps-track="post.click({ item: 2, priv: 0, post_type: 2 })" data-controller="se-share-sheet" data-se-share-sheet-title="Share a link to this answer" data-se-share-sheet-subtitle="" data-se-share-sheet-post-type="answer" data-se-share-sheet-social="facebook twitter " data-se-share-sheet-location="2" data-se-share-sheet-license-url="https%3a%2f%2fcreativecommons.org%2flicenses%2fby-sa%2f4.0%2f" data-se-share-sheet-license-name="CC BY-SA 4.0" data-s-popover-placement="bottom-start">Share</a> </div> <div class="flex--item"> <button type="button" class="js-cite-link s-btn s-btn__link">Cite</button> </div> <div class="flex--item"> <button type="button" id="btnFollowPost-3146487" class="s-btn s-btn__link js-follow-post js-follow-answer js-gps-track" data-gps-track="post.click({ item: 14, priv: 0, post_type: 2 })" data-controller="s-tooltip " data-s-tooltip-placement="bottom" data-s-popover-placement="bottom" aria-controls="" title="Follow this answer to receive notifications"> Follow <input type="hidden" id="voteFollowHash" value="69:3:31e,16:fa7b5ebd6422ec99,10:1740539661,16:5a27e774d9aaa177,7:3146487,e2c26dd8ab3c6b2c4c716f9908c98a93a79490cea1a6fa26170afc60a5a5b946" /> </button> </div> </div> <div class="js-menu-popup-container"></div> </div> </div> <div class="post-signature flex--item fl0"> <div class="user-info "> <div class="d-flex "> <div class="user-action-time fl-grow1"> <a href="/posts/3146487/revisions" title="show all edits to this post" class="js-gps-track" data-gps-track="post.click({ item: 4, priv: 0, post_type: 2 })">edited <span title='2019-05-08 14:17:22Z' class='relativetime'>May 8, 2019 at 14:17</span></a> </div> </div> <div class="user-gravatar32"> </div> <div class="user-details"> <div class="-flair"> </div> </div> </div> </div> <div class="post-signature flex--item fl0"> <div class="user-info user-hover "> <div class="d-flex "> <div class="user-action-time fl-grow1"> answered <span title='2019-03-13 11:51:52Z' class='relativetime'>Mar 13, 2019 at 11:51</span> </div> </div> <div class="user-gravatar32"> <a href="/users/637405/r29nb2xlifbp"><div class="gravatar-wrapper-32"><img src="https://i.sstatic.net/t1U95.png?s=64" alt="R29nb2xlIFBp's user avatar" width="32" height="32" class="bar-sm"></div></a> </div> <div class="user-details" itemprop="author" itemscope itemtype="http://schema.org/Person"> <a href="/users/637405/r29nb2xlifbp" dir="auto">R29nb2xlIFBp</a><span class="d-none" itemprop="name">R29nb2xlIFBp</span> <div class="-flair"> <span class="reputation-score" title="reputation score " dir="ltr">111</span><span title="6 bronze badges" aria-hidden="true"><span class="badge3"></span><span class="badgecount">6</span></span><span class="v-visible-sr">6 bronze badges</span> </div> </div> </div> </div> </div> </div> <span class="d-none">$\endgroup$</span> </div> <span class="d-none" itemprop="commentCount">3</span> <div class="post-layout--right js-post-comments-component"> <div id="comments-3146487" class="comments js-comments-container bt bc-black-200 mt12 " data-post-id="3146487" data-min-length="15"> <ul class="comments-list js-comments-list" data-remaining-comments-count="0" data-canpost="false" data-cansee="true" data-comments-unavailable="false" data-addlink-disabled="true"> <li id="comment-8221463" class="comment js-comment " data-comment-id="8221463" data-comment-owner-id="120442" data-comment-score="0"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-score js-comment-edit-hide"> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">Does this mean that, in the standard topology, every point in an open set is a limit point?</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <a href="/users/120442/ralph-dratman" title="111 reputation" class="comment-user">Ralph Dratman</a> </div> <span class="comment-date" dir="ltr"> <span class="v-visible-sr">Commented</span> <span title='2021-01-13 22:29:42Z, License: CC BY-SA 4.0' class='relativetime-clean'>Jan 13, 2021 at 22:29</span> </span> </div> </div> </li> <li id="comment-8221901" class="comment js-comment " data-comment-id="8221901" data-comment-owner-id="637405" data-comment-score="0"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-score js-comment-edit-hide"> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">Yes!Ralph . In point-set topology, every point of an open set is limit point and at the end of the day , what really makes the difference between an open and close set is exclusion and inclusion of boundary, respectively.</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <a href="/users/637405/r29nb2xlifbp" title="111 reputation" class="comment-user">R29nb2xlIFBp</a> </div> <span class="comment-date" dir="ltr"> <span class="v-visible-sr">Commented</span> <span title='2021-01-14 03:53:03Z, License: CC BY-SA 4.0' class='relativetime-clean'>Jan 14, 2021 at 3:53</span> </span> </div> </div> </li> <li id="comment-8224951" class="comment js-comment " data-comment-id="8224951" data-comment-owner-id="120442" data-comment-score="1"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-score js-comment-edit-hide"> <span title="number of 'useful comment' votes received" class="cool">1</span> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">Google Pi, thank you.</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <a href="/users/120442/ralph-dratman" title="111 reputation" class="comment-user">Ralph Dratman</a> </div> <span class="comment-date" dir="ltr"> <span class="v-visible-sr">Commented</span> <span title='2021-01-15 04:46:28Z, License: CC BY-SA 4.0' class='relativetime-clean'>Jan 15, 2021 at 4:46</span> </span> </div> </div> </li> </ul> </div> <div id="comments-link-3146487" data-rep=50 data-anon=true> <a class="js-add-link comments-link disabled-link" title="Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”." href="#" role="button">Add a comment</a> <span class="js-link-separator dno"> | </span> <a class="js-show-link comments-link dno" title="Expand to show all comments on this post" href=# onclick="" role="button"></a> </div> </div> </div> </div> <h2 class="bottom-notice"> You must <a href="/users/login?ssrc=question_page&returnurl=https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1290529">log in</a> to answer this question. </h2> </div> </div> <div id="sidebar" class="show-votes" role="complementary" aria-label="sidebar"> <div class="s-sidebarwidget s-sidebarwidget__yellow s-anchors s-anchors__grayscale mb16" data-tracker="cb=1"> <ul class="s-sidebarwidget--content s-sidebarwidget__items p0"> <li class="s-sidebarwidget--header"> Featured on Meta </li> <li class="s-sidebarwidget--item d-flex px16"> <div class="flex--item1 fl-shrink0"> <div class="favicon favicon-stackexchangemeta" title="Meta Stack Exchange"></div> </div> <div class="flex--item wmn0 ow-break-word"> <a href="https://meta.stackexchange.com/questions/406399/join-us-for-our-first-community-wide-ama-ask-me-anything-with-stack-overflow-s" class="js-gps-track" title="Join us for our first community-wide AMA (Ask Me Anything) with Stack Overflow’s CEO (and a few others) on February 26, 2025" data-ga="["community bulletin board","Featured on Meta","https://meta.stackexchange.com/questions/406399/join-us-for-our-first-community-wide-ama-ask-me-anything-with-stack-overflow-s",null,null]" data-gps-track="communitybulletin.click({ priority: 3, position: 0, location: questionpage })">Join us for our first community-wide AMA (Ask Me Anything) with Stack...</a> </div> </li> <li class="s-sidebarwidget--item d-flex px16"> <div class="flex--item1 fl-shrink0"> <div class="favicon favicon-stackexchangemeta" title="Meta Stack Exchange"></div> </div> <div class="flex--item wmn0 ow-break-word"> <a href="https://meta.stackexchange.com/questions/406575/bigbird-and-frog-have-joined-us-as-community-managers" class="js-gps-track" data-ga="["community bulletin board","Featured on Meta","https://meta.stackexchange.com/questions/406575/bigbird-and-frog-have-joined-us-as-community-managers",null,null]" data-gps-track="communitybulletin.click({ priority: 3, position: 1, location: questionpage })">bigbird and Frog have joined us as Community Managers</a> </div> </li> </ul> </div> <div class="js-zone-container zone-container-sidebar"> <div id="dfp-tsb" class="everyonelovesstackoverflow everyoneloves__top-sidebar"></div> <div class="js-report-ad-button-container " style="width: 300px"></div> </div> <div class="module sidebar-linked"> <h4 id="h-linked">Linked</h4> <div class="linked" data-tracker="lq=1"> <div class="spacer js-gps-track" data-gps-track="linkedquestion.click({ source_post_id: 1290529, target_question_id: 4796206, position: 0 })"> <a href="/q/4796206" title="Question score (upvotes - downvotes)"> <div class="answer-votes answered-accepted default">0</div> </a> <a href="/questions/4796206/clarification-on-boundary-and-limit-points-in-topology?noredirect=1" class="question-hyperlink">Clarification on Boundary and Limit Points in Topology</a> </div> </div> </div> <div class="module sidebar-related"> <h4 id="h-related">Related</h4> <div class="related js-gps-related-questions" data-tracker="rq=1"> <div class="spacer" data-question-id="374859"> <a href="/q/374859" title="Question score (upvotes - downvotes)" > <div class="answer-votes default">28</div> </a> <a href="/questions/374859/difference-between-limit-point-and-accumulation-point" class="question-hyperlink">Difference Between Limit Point and Accumulation Point?</a> </div> <div class="spacer" data-question-id="1071508"> <a href="/q/1071508" title="Question score (upvotes - downvotes)" > <div class="answer-votes answered-accepted default">13</div> </a> <a href="/questions/1071508/what-is-the-difference-between-the-limit-of-a-sequence-and-a-limit-point-of-a-se" class="question-hyperlink">What is the difference between the limit of a sequence and a limit point of a set?</a> </div> <div class="spacer" data-question-id="1856869"> <a href="/q/1856869" title="Question score (upvotes - downvotes)" > <div class="answer-votes default">0</div> </a> <a href="/questions/1856869/proving-that-only-two-points-of-intersection-are-sufficient-for-an-accumulation" class="question-hyperlink">Proving that only two points of intersection are sufficient for an accumulation point</a> </div> <div class="spacer" data-question-id="3039853"> <a href="/q/3039853" title="Question score (upvotes - downvotes)" > <div class="answer-votes answered-accepted default">0</div> </a> <a href="/questions/3039853/isolated-points-and-limit-points" class="question-hyperlink">Isolated points and limit points</a> </div> <div class="spacer" data-question-id="3719318"> <a href="/q/3719318" title="Question score (upvotes - downvotes)" > <div class="answer-votes default">0</div> </a> <a href="/questions/3719318/is-every-point-in-a-perfect-set-necessarily-a-condensation-point" class="question-hyperlink">Is every point in a perfect set necessarily a condensation point?</a> </div> <div class="spacer" data-question-id="4698513"> <a href="/q/4698513" title="Question score (upvotes - downvotes)" > <div class="answer-votes default">0</div> </a> <a href="/questions/4698513/definition-of-limit-points-of-a-set-vs-limit-point-of-a-sequence" class="question-hyperlink">definition of limit points of a set vs limit point of a sequence.</a> </div> <div class="spacer" data-question-id="5008065"> <a href="/q/5008065" title="Question score (upvotes - downvotes)" > <div class="answer-votes answered-accepted default">0</div> </a> <a href="/questions/5008065/different-definitions-of-limit-points" class="question-hyperlink">Different Definitions of Limit Points.</a> </div> </div> </div> <script type="text/javascript"> $(function() { $(".js-gps-related-questions .spacer").on("click", function () { fireRelatedEvent($(this).index() + 1, $(this).data('question-id')); }); function fireRelatedEvent(position, questionId) { StackExchange.using("gps", function() { StackExchange.gps.track('related_questions.click', { position: position, originQuestionId: 1290529, relatedQuestionId: +questionId, location: 'sidebar', source: 'Baseline' }); }); } }); </script> <div id="hot-network-questions" class="module tex2jax_ignore"> <h4> <a href="https://stackexchange.com/questions?tab=hot" class="js-gps-track s-link s-link__inherit" data-gps-track="posts_hot_network.click({ item_type:1, location:11 })"> Hot Network Questions </a> </h4> <ul> <li > <div class="favicon favicon-electronics" title="Electrical Engineering Stack Exchange"></div><a href="https://electronics.stackexchange.com/questions/739924/pre-amp-circuit-with-virtual-ground" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:135 }); posts_hot_network.click({ item_type:2, location:11 })"> Pre-amp circuit with virtual ground </a> </li> <li > <div class="favicon favicon-softwareengineering" title="Software Engineering Stack Exchange"></div><a href="https://softwareengineering.stackexchange.com/questions/456698/structuring-multiple-teams-within-an-organisation" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:131 }); posts_hot_network.click({ item_type:2, location:11 })"> Structuring multiple teams within an organisation </a> </li> <li > <div class="favicon favicon-ell" title="English Language Learners Stack Exchange"></div><a href="https://ell.stackexchange.com/questions/362513/is-it-correct-to-say-you-have-been-in-or-on-my-mind" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:481 }); posts_hot_network.click({ item_type:2, location:11 })"> Is it correct to say "you have been in or on my mind"? </a> </li> <li > <div class="favicon favicon-writing" title="Writing Stack Exchange"></div><a href="https://writing.stackexchange.com/questions/70894/where-to-learn-about-writing" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:166 }); posts_hot_network.click({ item_type:2, location:11 })"> Where to learn about writing? </a> </li> <li > <div class="favicon favicon-gis" title="Geographic Information Systems Stack Exchange"></div><a href="https://gis.stackexchange.com/questions/490671/improving-performance-and-index-use-on-postgis-query-with-st-intersects-and-st-w" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:79 }); posts_hot_network.click({ item_type:2, location:11 })"> Improving performance and index use on PostGIS query with ST_Intersects and ST_Within </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-superuser" title="Super User"></div><a href="https://superuser.com/questions/1883000/where-does-x-plore-cx-file-explorer-and-other-apps-mount-your-smb-path" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:3 }); posts_hot_network.click({ item_type:2, location:11 })"> Where does X-plore, Cx file Explorer and other apps mount your SMB (path)? </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-dsp" title="Signal Processing Stack Exchange"></div><a href="https://dsp.stackexchange.com/questions/96314/implementing-monopulse-tracking-in-python" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:295 }); posts_hot_network.click({ item_type:2, location:11 })"> Implementing Monopulse Tracking in python </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-mathoverflow" title="MathOverflow"></div><a href="https://mathoverflow.net/questions/488437/a-double-problem-on-compact-and-connected-sets-in-the-weak-topology" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:504 }); posts_hot_network.click({ item_type:2, location:11 })"> A double problem on compact and connected sets in the weak topology </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-rpg" title="Role-playing Games Stack Exchange"></div><a href="https://rpg.stackexchange.com/questions/214807/how-are-rounding-principles-chosen-in-system-design" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:122 }); posts_hot_network.click({ item_type:2, location:11 })"> How are rounding principles chosen in system design? </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-money" title="Personal Finance & Money Stack Exchange"></div><a href="https://money.stackexchange.com/questions/165389/could-i-miss-out-on-a-spike-if-i-sold-covered-call-options" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:93 }); posts_hot_network.click({ item_type:2, location:11 })"> Could I miss out on a spike if I sold covered call options? </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-scifi" title="Science Fiction & Fantasy Stack Exchange"></div><a href="https://scifi.stackexchange.com/questions/295154/what-songs-did-frankensteins-creature-react-to" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:186 }); posts_hot_network.click({ item_type:2, location:11 })"> What songs did Frankenstein's creature react to? </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-law" title="Law Stack Exchange"></div><a href="https://law.stackexchange.com/questions/107544/what-rule-of-canadian-law-would-allow-for-revocation-of-dual-citizenship-status" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:617 }); posts_hot_network.click({ item_type:2, location:11 })"> What rule of Canadian law would allow for revocation of dual-citizenship status and passport because of national interest/disloyalty etc.? </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-mechanics" title="Motor Vehicle Maintenance & Repair Stack Exchange"></div><a href="https://mechanics.stackexchange.com/questions/98662/how-to-tell-what-type-of-battery-my-car-has" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:224 }); posts_hot_network.click({ item_type:2, location:11 })"> How to tell what type of battery my car has? </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-physics" title="Physics Stack Exchange"></div><a href="https://physics.stackexchange.com/questions/843850/is-there-a-higgs-mechanism-for-higher-form-symmetries" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:151 }); posts_hot_network.click({ item_type:2, location:11 })"> Is there a Higgs mechanism for higher-form symmetries? </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-space" title="Space Exploration Stack Exchange"></div><a href="https://space.stackexchange.com/questions/67962/uncrewed-lunar-probe-transit-times-space-age-era-vs-contemporary" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:508 }); posts_hot_network.click({ item_type:2, location:11 })"> Uncrewed Lunar Probe Transit Times.....Space Age Era vs Contemporary </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-electronics" title="Electrical Engineering Stack Exchange"></div><a href="https://electronics.stackexchange.com/questions/739919/please-advise-if-this-led-circuit-is-efficient" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:135 }); posts_hot_network.click({ item_type:2, location:11 })"> Please advise if this LED circuit is efficient </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-unix" title="Unix & Linux Stack Exchange"></div><a href="https://unix.stackexchange.com/questions/791589/ssh-access-broken-probably-due-to-my-dislike-of-networkmanager" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:106 }); posts_hot_network.click({ item_type:2, location:11 })"> SSH access broken; probably due to my dislike of 'NetworkManager' </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-mathoverflow" title="MathOverflow"></div><a href="https://mathoverflow.net/questions/488452/rational-independence-of-square-roots-of-polynomials" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:504 }); posts_hot_network.click({ item_type:2, location:11 })"> Rational independence of square roots of polynomials </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-mattermodeling" title="Matter Modeling Stack Exchange"></div><a href="https://mattermodeling.stackexchange.com/questions/14030/what-is-the-conversion-factor-from-ev-atom-to-kj-mol" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:704 }); posts_hot_network.click({ item_type:2, location:11 })"> What is the conversion factor from eV/atom to kJ/mol? </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-academia" title="Academia Stack Exchange"></div><a href="https://academia.stackexchange.com/questions/217024/how-does-a-postdoc-differ-from-a-phd" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:415 }); posts_hot_network.click({ item_type:2, location:11 })"> How does a PostDoc differ from a PhD? </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-philosophy" title="Philosophy Stack Exchange"></div><a href="https://philosophy.stackexchange.com/questions/122994/can-truth-exist-without-language" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:265 }); posts_hot_network.click({ item_type:2, location:11 })"> Can truth exist without language? </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-tex" title="TeX - LaTeX Stack Exchange"></div><a href="https://tex.stackexchange.com/questions/737862/mathgreekupright-inconsistent-behavior-in-latest-context" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:85 }); posts_hot_network.click({ item_type:2, location:11 })"> `\mathgreekupright` inconsistent behavior in latest ConTeXt </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-physics" title="Physics Stack Exchange"></div><a href="https://physics.stackexchange.com/questions/843868/how-does-the-schwarzschild-metric-account-for-time-dilation-in-the-vicinity-of-a" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:151 }); posts_hot_network.click({ item_type:2, location:11 })"> How does the Schwarzschild metric account for time dilation in the vicinity of a massive object? </a> </li> <li class="dno js-hidden"> <div class="favicon favicon-unix" title="Unix & Linux Stack Exchange"></div><a href="https://unix.stackexchange.com/questions/791622/run-command-on-each-line-of-csv-file-using-fields-in-different-places-of-the-co" class="js-gps-track question-hyperlink mb0" data-gps-track="site.switch({ item_type:11, target_site:106 }); posts_hot_network.click({ item_type:2, location:11 })"> Run command on each line of CSV file, using fields in different places of the command </a> </li> </ul> <a href="#" class="show-more js-show-more js-gps-track" data-gps-track="posts_hot_network.click({ item_type:3, location:11 })"> more hot questions </a> </div> <div id="feed-link" class="js-feed-link"> <a href="/feeds/question/1290529" title="Feed of this question and its answers"> <svg aria-hidden="true" class="fc-orange-400 svg-icon iconRss" width="18" height="18" viewBox="0 0 18 18"><path d="M3 1a2 2 0 0 0-2 2v12c0 1.1.9 2 2 2h12a2 2 0 0 0 2-2V3a2 2 0 0 0-2-2zm0 1.5c6.9 0 12.5 5.6 12.5 12.5H13C13 9.55 8.45 5 3 5zm0 5c4.09 0 7.5 3.41 7.5 7.5H8c0-2.72-2.28-5-5-5zm0 5c1.36 0 2.5 1.14 2.5 2.5H3z"/></svg> Question feed </a> </div> <aside class="s-modal js-feed-link-modal" tabindex="-1" role="dialog" aria-labelledby="feed-modal-title" aria-describedby="feed-modal-description" aria-hidden="true"> <div class="s-modal--dialog js-modal-dialog wmx4" role="document" data-controller="se-draggable"> <h1 class="s-modal--header fw-bold js-first-tabbable" id="feed-modal-title" data-se-draggable-target="handle" tabindex="0"> Subscribe to RSS </h1> <div class="d-flex gs4 gsy fd-column"> <div class="flex--item"> <label class="d-block s-label c-default" for="feed-url"> Question feed <p class="s-description mt2" id="feed-modal-description">To subscribe to this RSS feed, copy and paste this URL into your RSS reader.</p> </label> </div> <div class="d-flex ps-relative"> <input class="s-input" type="text" name="feed-url" id="feed-url" readonly="readonly" value="https://math.stackexchange.com/feeds/question/1290529" /> <svg aria-hidden="true" class="s-input-icon fc-orange-400 svg-icon iconRss" width="18" height="18" viewBox="0 0 18 18"><path d="M3 1a2 2 0 0 0-2 2v12c0 1.1.9 2 2 2h12a2 2 0 0 0 2-2V3a2 2 0 0 0-2-2zm0 1.5c6.9 0 12.5 5.6 12.5 12.5H13C13 9.55 8.45 5 3 5zm0 5c4.09 0 7.5 3.41 7.5 7.5H8c0-2.72-2.28-5-5-5zm0 5c1.36 0 2.5 1.14 2.5 2.5H3z"/></svg> </div> </div> <a class="s-modal--close s-btn s-btn__muted js-modal-close js-last-tabbable" href="#" aria-label="Close"> <svg aria-hidden="true" class="svg-icon iconClearSm" width="14" height="14" viewBox="0 0 14 14"><path d="M12 3.41 10.59 2 7 5.59 3.41 2 2 3.41 5.59 7 2 10.59 3.41 12 7 8.41 10.59 12 12 10.59 8.41 7z"/></svg> </a> </div> </aside> </div> </div> <script>StackExchange.ready(function(){$.get('/posts/1290529/ivc/e9f2?prg=b3345b91-21be-4181-9c28-9927a7e52e11');});</script> <noscript><div><img src="/posts/1290529/ivc/e9f2?prg=b3345b91-21be-4181-9c28-9927a7e52e11" class="dno" alt="" width="0" height="0"></div></noscript></div> <script defer src="https://cdn.sstatic.net/Js/questions/show.en.js?v=03feb8d88a8b"></script> <script type="application/json" data-role="module-args" data-module-name="entry-points/questions/show.mod">{"doMarkObsolete":false,"doTimeOnPage":false,"sendAnswerEditorEventsToAEH":false,"sendAnswerEditorEventsToPrizm":false,"questionId":1290529,"sendAuthQuestionVisitToAEH":false,"sendAuthQuestionVisitToPrizm":false}</script> </div> </div> <script type="text/javascript"> var cam = cam || { opt: {} }; var clcGamLoaderOptions = cam || { opt: {} }; var opt = clcGamLoaderOptions.opt; opt.refresh = !1; opt.refreshInterval = 0; opt.sf = !1; opt.hb = !1; opt.ll = !0; opt.tlb_position = 0; opt.personalization_consent = !1; opt.targeting_consent = !1; opt.performance_consent = !1; opt.targeting = {Registered:['false'],Reputation:['new'],NumberOfAnswers:['3'],cf_bot_score:'1'}; opt.adReportEnabled = !1; opt.adReportUrl = '/ads/report-ad'; opt.adReportText = 'Report this ad'; opt.adReportFileTypeErrorMessage = 'Please select a PNG or JPG file.'; opt.adReportFileSizeErrorMessage = 'The file must be under 2 MiB.'; opt.adReportErrorText = 'Error uploading ad report.'; opt.adReportThanksText = 'Thanks for your feedback. We’ll review this against our code of conduct and take action if necessary.'; opt.adReportLoginExpiredMessage = 'Your login session has expired, please login and try again.'; opt.adReportLoginErrorMessage = 'An error occurred when loading the report form - please try again'; opt.adReportModalClass = 'js-ad-report'; opt.countryCode = 'SG'; opt.qualtricsSurveyData = '{"isRegistered":"False","repBucket":"new","referrer":"https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1290529%2fdifference-between-boundary-point-limit-point%2f1290541","accountAge":"0"}'; opt.perRequestGuid = 'b3345b91-21be-4181-9c28-9927a7e52e11'; opt.responseHash = 'b7hZC/ej51CiSVPJKgWYTuwIdNCp3BqKp83lJg5vf5o='; opt.targeting.TargetingConsent = ['False_Passive']; opt.allowAccountTargetingForThisRequest = !1; const urlParams = new URLSearchParams(window.location.search); if (urlParams.has('dfptestads')) { const dfptestads = urlParams.get('dfptestads'); opt.targeting.DfpTestAds = dfptestads; } </script> <script>;(()=>{"use strict";var __webpack_modules__={23:(e,t,s)=>{s.d(t,{Z7:()=>c,eq:()=>l,kG:()=>d});const n="248424177",o=(a=location.pathname,/^\/tags\//.test(a)||/^\/questions\/tagged\//.test(a)?"tag-pages":/^\/discussions\//.test(a)||/^\/beta\/discussions/.test(a)?"discussions":/^\/$/.test(a)||/^\/home/.test(a)?"home-page":/^\/jobs$/.test(a)||/^\/jobs\//.test(a)?"jobs":"question-pages");var a;let i=location.hostname;const r={slots:{lb:[[728,90]],mlb:[[728,90]],smlb:[[728,90]],bmlb:[[728,90]],sb:e=>"dfp-tsb"===e?[[300,250],[300,600]]:[[300,250]],"tag-sponsorship":[[730,135]],"mobile-below-question":[[320,50],[300,250]],msb:[[300,250],[300,600]],"talent-conversion-tracking":[[1,1]],"site-sponsorship":[[230,60]]},ids:{"dfp-tlb":"lb","dfp-mlb":"mlb","dfp-smlb":"smlb","dfp-bmlb":"bmlb","dfp-tsb":"sb","dfp-isb":"sb","dfp-tag":"tag-sponsorship","dfp-msb":"msb","dfp-sspon":"site-sponsorship","dfp-m-aq":"mobile-below-question"},idsToExcludeFromAdReports:["dfp-sspon"]};function d(){return Object.keys(r.ids)}function l(e){return r.idsToExcludeFromAdReports.indexOf(e)<0}function c(e,t){var s=e.split("_")[0];const a=r.ids[s];let d=r.slots[a];return"function"==typeof d&&(d=d(s)),{path:`/${n}/${t||i}/${a}/${o}`,sizes:d,zone:a}}},865:(e,t,s)=>{function n(e){return"string"==typeof e?document.getElementById(e):e}function o(e){return!!(e=n(e))&&"none"===getComputedStyle(e).display}function a(e){return!o(e)}function i(e){return!!e}function r(e){return/^\s*$/.test(n(e).innerHTML)}function d(e){const{style:t}=e;t.height=t.maxHeight=t.minHeight="auto",t.display="none"}function l(e){const{style:t}=e;t.height=t.maxHeight=t.minHeight="auto",t.display="none",[].forEach.call(e.children,l)}function c(e){const{style:t}=e;t.height=t.maxHeight=t.minHeight="auto",t.removeProperty("display")}function g(e){const t=document.createElement("script");t.src=e,document.body.appendChild(t)}function p(e){return s=e,(t=[]).push=function(e){return s(),delete this.push,this.push(e)},t;var t,s}function h(e){let t="function"==typeof HTMLTemplateElement;var s=document.createElement(t?"template":"div");return e=e.trim(),s.innerHTML=e,t?s.content.firstChild:s.firstChild}s.d(t,{$Z:()=>c,Bv:()=>h,Gx:()=>g,Nj:()=>n,QZ:()=>p,cf:()=>d,pn:()=>a,wo:()=>l,xb:()=>r,xj:()=>o,yb:()=>i})},763:(__unused_webpack_module,__webpack_exports__,__webpack_require__)=>{__webpack_require__.d(__webpack_exports__,{t:()=>AdReports});var _common_helper__WEBPACK_IMPORTED_MODULE_2__=__webpack_require__(865),_console__WEBPACK_IMPORTED_MODULE_1__=__webpack_require__(276),_ad_units__WEBPACK_IMPORTED_MODULE_0__=__webpack_require__(23);class AdReports{constructor(e,t){if(this.googletag=e,this.cam=t,this.allowedFileTypes=["image/png","image/jpg","image/jpeg"],this.ignoreValidation=!1,_console__WEBPACK_IMPORTED_MODULE_1__.cM("Ad reporting init"),this.cam=t,this.callOnButtonClick=e=>this.onButtonClick(e),this.googletag.pubads().addEventListener("slotRenderEnded",e=>this.handleSlotRendered(e)),Array.isArray(t.slotsRenderedEvents)){_console__WEBPACK_IMPORTED_MODULE_1__.cM("Adding report button to "+t.slotsRenderedEvents.length+" events that have transpired");for(var s=0;s<t.slotsRenderedEvents.length;s++)this.handleSlotRendered(t.slotsRenderedEvents[s])}}handleSlotRendered(e){if(e&&e.slot&&!e.isEmpty&&(e.creativeId||e.lineItemId||!e.isEmpty)){var t=e.slot.getSlotElementId();if(t){var s=document.getElementById(t);if(s)if((0,_ad_units__WEBPACK_IMPORTED_MODULE_0__.eq)(t)){var n=s?.closest(".js-zone-container")?.querySelector(".js-report-ad-button-container");n?(n.innerHTML="",n.append(this.createButton(e)),n.style.height="24px",_console__WEBPACK_IMPORTED_MODULE_1__.cM("Added report button to the bottom of "+t)):_console__WEBPACK_IMPORTED_MODULE_1__.cM("Ad report button not found, may be intentional, element: "+t)}else _console__WEBPACK_IMPORTED_MODULE_1__.cM("Not adding report button to the bottom of "+t+": shouldHaveReportButton = false");else _console__WEBPACK_IMPORTED_MODULE_1__.cM("Not adding report button to the bottom of "+t+": resolved invalid adUnit element")}else _console__WEBPACK_IMPORTED_MODULE_1__.cM("Not adding report button to the bottom of element: invalid adUnitElementId")}else _console__WEBPACK_IMPORTED_MODULE_1__.cM("Not adding report button to the bottom of element: invalid SlotRenderEndedEvent")}async onButtonClick(e){e.preventDefault();let t=e.target;const s=t.dataset.modalUrl,n=t.dataset.googleEventData;return await this.loadModal(s,t,n),!1}createButton(e){let t=document.createElement("button");var s=JSON.stringify(e);return t.dataset.googleEventData=s,t.dataset.modalUrl=this.cam.opt.adReportUrl,t.dataset.adUnit=e.slot.getSlotElementId(),t.classList.add("js-report-ad","s-btn","s-btn__link","fs-fine","mt2","float-right"),t.append(document.createTextNode(this.cam.opt.adReportText)),t.removeEventListener("click",this.callOnButtonClick),t.addEventListener("click",this.callOnButtonClick),t}async loadModal(url,$link,googleEventData){try{await window.StackExchange.helpers.loadModal(url,{returnElements:window.$($link)}),this.initForm(googleEventData)}catch(e){var message="",response=e.responseText?eval(`(${e.responseText})`):null;message=response&&response.isLoggedOut?this.cam.opt.adReportLoginExpiredMessage:this.cam.opt.adReportLoginErrorMessage,window.StackExchange.helpers.showToast(message,{type:"danger"})}}removeModal(){window.StackExchange.helpers.closePopups(document.querySelectorAll("."+this.cam.opt.adReportModalClass),"dismiss")}initForm(e,t=!1){this.ignoreValidation=t,this.$form=document.querySelector(".js-ad-report-form"),this.$googleEventData=this.$form.querySelector(".js-json-data"),this.$adReportReasons=this.$form.querySelectorAll(".js-ad-report-reason"),this.$adReportReasonOther=this.$form.querySelector(".js-ad-report-reason-other"),this.$fileUploaderInput=this.$form.querySelector(".js-file-uploader-input"),this.$imageUploader=this.$form.querySelector(".js-image-uploader"),this.$clearImageUpload=this.$form.querySelector(".js-clear-image-upload"),this.$imageUploaderText=this.$form.querySelector(".js-image-uploader-text"),this.$imageUploaderPreview=this.$form.querySelector(".js-image-uploader-preview"),this.$fileErrorMessage=this.$form.querySelector(".js-file-error");const s=this.$form.querySelector(".js-drag-drop-enabled"),n=this.$form.querySelector(".js-drag-drop-disabled");this.$googleEventData.value=e,this.$adReportReasons.forEach((e,t)=>e.addEventListener("change",e=>{this.$adReportReasonOther.classList.toggle("d-none","3"!==e.target.value)})),this.$fileUploaderInput.addEventListener("change",()=>{this.validateFileInput()&&this.updateImagePreview(this.$fileUploaderInput.files)}),this.$clearImageUpload.addEventListener("click",e=>{e.preventDefault(),this.clearImageUpload()});try{this.$fileUploaderInput[0].value="",this.$imageUploader.addEventListener("dragenter dragover dragleave drop",this.preventDefaults),this.$imageUploader.addEventListener("dragenter dragover",this.handleDragStart),this.$imageUploader.addEventListener("dragleave drop",this.handleDragEnd),this.$imageUploader.addEventListener("drop",this.handleDrop)}catch(e){s.classList.add("d-none"),n.classList.remove("d-none")}this.$form.removeEventListener("",this.handleDragEnd),this.$form.addEventListener("submit",async e=>(e.preventDefault(),this.submitForm(),!1))}clearImageUpload(){this.$fileUploaderInput.value="",this.$imageUploaderPreview.setAttribute("src",""),this.$imageUploaderPreview.classList.add("d-none"),this.$clearImageUpload.classList.add("d-none"),this.$imageUploaderText.classList.remove("d-none"),this.$imageUploader.classList.add("p16","ba","bas-dashed","bc-black-100")}preventDefaults(e){e.preventDefault(),e.stopPropagation()}handleDragStart(e){this.$imageUploader.classList.remove("bas-dashed"),this.$imageUploader.classList.add("bas-solid","bc-black-100")}handleDragEnd(e){this.$imageUploader.classList.remove("bas-solid","bc-black-100"),this.$imageUploader.classList.add("bas-dashed")}handleDrop(e){var t=e.originalEvent.dataTransfer.files;FileReader&&t&&1===t.length&&(this.$fileUploaderInput.files=t,this.validateFileInput()&&this.updateImagePreview(t))}setError(e){this.$fileErrorMessage.parentElement.classList.toggle("has-error",e)}updateImagePreview(e){this.$imageUploader.classList.remove("p16","ba","bas-dashed","bc-black-100"),this.$clearImageUpload.classList.remove("d-none"),this.$imageUploaderText.classList.add("d-none");var t=new FileReader;t.onload=e=>{null!=e.target&&(this.$imageUploaderPreview.setAttribute("src",e.target.result),this.$imageUploaderPreview.classList.remove("d-none"))},t.readAsDataURL(e[0])}validateFileInput(){if(this.ignoreValidation)return!0;const e=this.cam.opt.adReportFileTypeErrorMessage,t=this.cam.opt.adReportFileSizeErrorMessage;if(null==this.$fileUploaderInput.files)return!1;var s=this.$fileUploaderInput.files[0];return null==s?(this.setError(!0),!1):this.allowedFileTypes.indexOf(s.type)<0?(this.$fileErrorMessage.textContent=e,this.$fileErrorMessage.classList.remove("d-none"),this.setError(!0),!1):s.size>2097152?(this.$fileErrorMessage.textContent=t,this.$fileErrorMessage.classList.remove("d-none"),this.setError(!0),!1):(this.$fileErrorMessage.classList.add("d-none"),this.setError(!1),!0)}async gatherDiagnosticInfo(){return{BrowserVersion:await this.getBrowserVersion()}}getElementSource(e){return e.outerHTML}getNestedIFrameElement(e){var t=e.querySelector("iframe");return t.contentDocument?t.contentDocument.documentElement:t.contentWindow.document.documentElement}async getBrowserVersion(){return await navigator.userAgentData.getHighEntropyValues(["fullVersionList"]).then(e=>JSON.stringify(e.fullVersionList))}async submitForm(){if(!this.validateFileInput())return!1;this.$form.querySelector("[type=submit]").setAttribute("disabled","true");var e=JSON.parse(this.$googleEventData.value||"{}");e.Reason=parseInt(this.$form.querySelector(".js-ad-report-reason:checked").value,10),e.Description=this.$adReportReasonOther.value,this.$googleEventData.value=JSON.stringify(e);var t=new FormData(this.$form);if("1"===t.get("shareDiagnosticInfo")){var s=await this.gatherDiagnosticInfo();Object.keys(s).forEach(e=>t.append(e,s[e]))}try{const e=await window.fetch(this.$form.getAttribute("action"),{method:this.$form.getAttribute("method"),body:t,cache:"no-cache"}),s=e.headers.get("content-type")||"",o=await e.text();if(!e.ok)throw new Error("response not valid");if(0===s.indexOf("text/html")){var n=(0,_common_helper__WEBPACK_IMPORTED_MODULE_2__.Bv)(o);const e=n?n.querySelector(".js-modal-content"):null;if(_console__WEBPACK_IMPORTED_MODULE_1__.cM("$popupContent"),_console__WEBPACK_IMPORTED_MODULE_1__.cM(e),!e)throw new Error(`Could not find .js-modal-content in response from ${this.$form.getAttribute("action")}`);document.querySelector(".js-modal-content").replaceWith(e)}else window.StackExchange.helpers.showToast(this.cam.opt.adReportThanksText,{type:"success"}),this.removeModal()}catch(e){window.StackExchange.helpers.showToast(this.cam.opt.adReportErrorText,{type:"danger"})}finally{let e=this.$form.querySelector("[type=submit]");e&&e.removeAttribute("disabled")}}}},276:(e,t,s)=>{function n(...e){}function o(...e){}s.d(t,{cM:()=>n,vU:()=>o})}},__webpack_module_cache__={};function __webpack_require__(e){var t=__webpack_module_cache__[e];if(void 0!==t)return t.exports;var s=__webpack_module_cache__[e]={exports:{}};return __webpack_modules__[e](s,s.exports,__webpack_require__),s.exports}__webpack_require__.d=(e,t)=>{for(var s in t)__webpack_require__.o(t,s)&&!__webpack_require__.o(e,s)&&Object.defineProperty(e,s,{enumerable:!0,get:t[s]})},__webpack_require__.o=(e,t)=>Object.prototype.hasOwnProperty.call(e,t);var __webpack_exports__={};(()=>{var e=__webpack_require__(276),t=(e=>(e[e.Above=0]="Above",e[e.Below=1]="Below",e))(t||{});const s=Object.assign({},{"lib":"https://clc.stackoverflow.com/Content/bundles/js/gam_loader_script.bundle.741.5deb8dc1a671bfb1646b.js?v=7178a92a336c","style":null,"u":null,"wa":true,"kt":2000,"tto":true,"h":"clc.stackoverflow.com","allowed":"^(((talent\\.)?stackoverflow)|(blog\\.codinghorror)|(.*\\.googlesyndication)|(serverfault|askubuntu|superuser)|([^\\.]+\\.stackexchange))\\.com$","wv":true,"al":false,"abd":true,"cpa_liid":[5882654614],"cpa_cid":[138377597667],"dp":false,"tgt_to":1000,"tgt_u":"https://clc.stackoverflow.com/get-user-acct-tgt","tgt_e":true,"tgt_p":100,"dv_enabled":false,"upo":""});var n=__webpack_require__(23),o=__webpack_require__(865),a=__webpack_require__(763);class i{constructor(t,s){this.googletag=t,this.interval=s,e.cM("Ad refresh init. interval: "+s),this.googletag.pubads().addEventListener("impressionViewable",e=>this.onImpressionViewable(e)),e.cM("done enabling ad refresh")}onImpressionViewable(t){var s=t.slot;e.cM("ad refresh - slot "+s.getSlotElementId()+" is viewable, initializing refresh"),this.scheduleRefresh(s)}scheduleRefresh(e){setTimeout(()=>this.refreshAdSlot(e),1e3*this.interval)}static refreshMyAd(t,s){let n=t.pubads().getSlots().find(e=>e.getSlotElementId()===s);n&&(e.cM("refreshMyAd - refreshing ad slot "+s),t.pubads().refresh([n]))}static removeMyAd(t,s){let n=t.pubads().getSlots().find(e=>e.getSlotElementId()===s);n&&(e.cM("removeMyAd - destroying ad slot "+s),t.destroySlots([n]))}refreshAdSlot(t){var s=t.getSlotElementId();this.isElementVisibleInBrowser(s)?(e.cM("refreshing ad slot "+s),googletag.pubads().refresh([t])):(e.cM("refresh skipped this time; ad slot not viewable:"+s),this.scheduleRefresh(t))}isElementVisibleInBrowser(e){var t=document.getElementById(e);if(null!==t){var s=t.getBoundingClientRect();if(s.top>=0&&s.left>=0&&s.bottom<=(window.innerHeight||document.documentElement.clientHeight)&&s.right<=(window.innerWidth||document.documentElement.clientWidth))return!0}return!1}}var r=(e=>(e.Off="Off",e.PreSurvey="PreSurvey",e.Collect="Collect",e.PostSurvey="PostSurvey",e))(r||{});class d{constructor(e,t){this.lineItemImpressions=[],this.surveysIdsCompleted=[],this.lineItemImpressions=e,this.surveysIdsCompleted=t}addImpression(e,t){let s={brandId:e,lineItemId:t,timestamp:new Date};this.lineItemImpressions.push(s)}addBrandSurveyCompleted(e){-1===this.surveysIdsCompleted.indexOf(e)&&this.surveysIdsCompleted.push(e)}getTotalBrandImpressions(){let e=new Map;for(let t of this.lineItemImpressions)if(e.has(t.brandId)){let s=e.get(t.brandId);e.set(t.brandId,s+1)}else e.set(t.brandId,1);return e}getBrandLineItemImpressions(e){let t={};for(let s of this.lineItemImpressions)if(s.brandId==e)if(void 0!==t[s.lineItemId]){let e=t[s.lineItemId];t[s.lineItemId]=e+1}else t[s.lineItemId]=1;return t}}class l{constructor(){this.surveyEngagementLocalStorageKey="clc-survey-engagement"}getBrandSurveyEngagement(){let e=localStorage.getItem(this.surveyEngagementLocalStorageKey);if(null===e)return new d([],[]);let t=JSON.parse(e);return new d(t.lineItemImpressions,t.surveysIdsCompleted)}saveBrandSurveyEngagement(e){let t=JSON.stringify(e);localStorage.setItem(this.surveyEngagementLocalStorageKey,t)}}class c{constructor(){this.surveyRepository=new l}getBrandSurveyEngagement(){return this.surveyRepository.getBrandSurveyEngagement()}recordImpression(e,t){let s=this.getBrandSurveyEngagement();s.addImpression(e,t),this.surveyRepository.saveBrandSurveyEngagement(s)}recordBrandSurveyCompleted(e){let t=this.getBrandSurveyEngagement();t.addBrandSurveyCompleted(e),this.surveyRepository.saveBrandSurveyEngagement(t)}}class g{constructor(t,s){this.googletag=t,this.brandSettings=s,this.brandSlotMap=new Map,this.brandSurveyEngagementService=new c,e.cM("Brand Survey init: "+JSON.stringify(s)),void 0!==s?(this.googletag.pubads().addEventListener("slotRenderEnded",e=>this.handleSlotRendered(e)),this.googletag.pubads().addEventListener("impressionViewable",e=>this.onImpressionViewable(e)),e.cM("done enabling Brand Survey")):e.cM("Brand Survey init: brandSettings is undefined, not initializing")}handleSlotRendered(t){e.cM("Brand Survey - slot rendered - slot:"+JSON.stringify(t.slot.getSlotElementId())+" lineItem: "+t.lineItemId);let s=this.findItemWithId(t.lineItemId);if(null===s||s.mode!==r.Collect)this.brandSlotMap.delete(t.slot.getSlotElementId());else{let e={brandId:s.brandId,lineItemId:t.lineItemId};this.brandSlotMap.set(t.slot.getSlotElementId(),e)}}onImpressionViewable(t){let s=t.slot;if(e.cM("ad - Brand Survey - impression viewable. Details: "+JSON.stringify(s.getSlotElementId())),e.cM("ad - Brand Survey - slot "+s.getSlotElementId()+" is viewable"),this.brandSlotMap.has(s.getSlotElementId())){let t=this.brandSlotMap.get(s.getSlotElementId());e.cM("Brand Survey - brand "+t.brandId+" is viewable"),this.recordImpression(this.brandSlotMap.get(s.getSlotElementId()))}}recordImpression(t){e.cM("ad - Brand Survey - recording impression for brand "+t.brandId),this.brandSurveyEngagementService.recordImpression(t.brandId,t.lineItemId)}findItemWithId(t){return e.cM("brand settings: "+JSON.stringify(this.brandSettings)),this.brandSettings.find(e=>e.lineItemIds.includes(t))||null}}const p="response-brand-survey-submit|",h="request-brand-survey-metadata|",m="record-metric-on-server|",u="request-dsp-tags",f="response-dsp-tags|";class _{static refreshAdIfBrandSurveyIsDuplicated(e,t,s){if(this.alreadyCompletedThisBrandSurvey(t)){var n=document.getElementById(s).closest(".js-zone-container");i.removeMyAd(e,s),n&&n.remove()}}static alreadyCompletedThisBrandSurvey(e){return(new c).getBrandSurveyEngagement().surveysIdsCompleted.includes(e)}}window.cam=new class{constructor(t=null){if(this.gptImported=!1,this.slotsRenderedEvents=[],this.collapsed={},e.cM("constructor"),this.clc_options=s,window.clcGamLoaderOptions)Object.assign(this,window.clcGamLoaderOptions);else if(void 0===this.opt){let e=window.opt;e&&(this.opt=e)}}init(){if(e.cM("init"),void 0===this.opt)throw new Error("opt not set, required by GAM Loader");e.cM("init brand survey service"),this.getUserMetaPromise=this.getUserMeta(),e.cM("setup message handler"),window.addEventListener("message",e=>{this.onmessage(e)})}handleSlotRenderedNoAdReport(){if(googletag.pubads().addEventListener("slotRenderEnded",e=>this.applyExtraMarginBottom(e)),Array.isArray(this.slotsRenderedEvents))for(var e=0;e<this.slotsRenderedEvents.length;e++)this.applyExtraMarginBottom(this.slotsRenderedEvents[e])}onmessage(t){let s="omni";if(t.data&&("string"==typeof t.data||t.data instanceof String))if(0===t.data.indexOf("get-omni-")){e.cM("Recevied get-omni message, sending back omni");var n=t.source,a=this.opt.omni,i="string"==typeof a?a:"";n.postMessage([s,i,this.opt.perRequestGuid].join("|"),"*")}else if(0===t.data.indexOf("collapse-")){e.cM("Recevied collapse message, collapse ad iframe"),e.cM(t);for(var r=t.source.window,d=document.getElementsByTagName("IFRAME"),l=0;l<d.length;l++){var g=d[l];if(g.contentWindow==r)return void(0,o.wo)(g.parentElement.parentElement.parentElement)}}else if(0===t.data.indexOf("resize|")){e.cM("Recevied resize message, resize ad iframe"),e.cM(t);let s=this._getFrameByEvent(t),n=t.data.indexOf("|")+1,o=t.data.slice(n),a=parseFloat(o)+.5;e.cM("New iframe height "+a),s.height=a.toString(),s.parentElement.style.height=a.toString()+"px"}else if(0===t.data.indexOf("getmarkup|")){let s=t.data.indexOf("|")+1,n=t.data.slice(s);e.cM("Recevied get markup message: "+n);let o=this._getFrameByEvent(t).closest(".everyonelovesstackoverflow");const a=document.createElement("script");a.dataset.adZoneId=o.id,a.src=n,document.body.appendChild(a)}else if(0===t.data.indexOf("window-location|")){let s=t.data.indexOf("|")+1,n=t.data.slice(s);e.cM("Recevied window location message: "+n),n.startsWith("/")||(n="/"+n),window.open(window.location.protocol+"//"+window.location.host+n,"_blank")}else if(0===t.data.indexOf("request-brand-survey-submit|")){let s=t.data.split("|"),n=s[1],o=s[2],a=s[3],i=JSON.parse(a);e.cM(n),e.cM(o),e.cM(a),e.cM("Received brand survey "+n+" response message: "+o);var v=new FormData;for(var b in i)v.append(b,i[b]);let r=this._getFrameByEvent(t);if(_.alreadyCompletedThisBrandSurvey(+n))return e.cM("Already completed this brand survey. Not submitting duplicate to server."),void r.contentWindow.postMessage("response-brand-survey-submit-duplicate|","*");e.cM("Send the brand survey to the server"),fetch(o,{method:"POST",body:v}).then(e=>e.json()).then(e=>r.contentWindow.postMessage({messageType:p},"*")).catch(e=>r.contentWindow.postMessage({messageType:p},"*"))}else if(0===t.data.indexOf("brand-survey-completed-store|")){let s=t.data.split("|"),n=(s[1],s[2]);if(e.cM("Received brand survey completed store message for survey ID "+n),_.alreadyCompletedThisBrandSurvey(+n))return void e.cM("Already completed this brand survey. Not recording duplicate locally.");e.cM("Record brand survey completion locally"),(new c).recordBrandSurveyCompleted(+n)}else if(0===t.data.indexOf(h)){let s=t.data.split("|"),n=s[1],o=s[2];e.cM("Received message: "+h+" with Brand Survey ID "+o);let a=(new c).getBrandSurveyEngagement().getBrandLineItemImpressions(+n),i=JSON.stringify(a),r=this._getFrameByEvent(t);e.cM("sending impression data: "+i),r.contentWindow.postMessage("response-brand-survey-metadata|"+this.opt.responseHash+"|"+this.opt.perRequestGuid+"|"+i+"|"+this.opt.countryCode+"|"+this.opt.qualtricsSurveyData,"*")}else if(0===t.data.indexOf("refresh-if-duplicate-brand-survey|")){let e=t.data.split("|")[1],s=this.getSlotElementIdByEvent(t);_.refreshAdIfBrandSurveyIsDuplicated(googletag,+e,s)}else if(0===t.data.indexOf(m)){e.cM("Received message: "+m+" with args: "+t.data);let s=t.data.split("|"),n=s[1],o=s[2],a=s[3],i=s[4],r=new FormData;r.append("brandSurveyId",a.toString()),r.append("responseHash",this.opt.responseHash),r.append("perRequestGuid",this.opt.perRequestGuid),r.append("questionNumber",n.toString()),r.append("metricType",i.toString()),fetch(o,{method:"POST",body:r}).then(e=>e.ok).catch(t=>{e.cM("SendMetricToServer: Error sending metric to server: "+t)})}else if(0===t.data.indexOf(u)){e.cM("Received message: "+u+" with args: "+t.data);let s=this._getFrameByEvent(t);if(!this.opt.targeting["so-tag"])return void s.contentWindow.postMessage(f,"*");const n=this.opt.targeting["so-tag"].join(",");e.cM("sending targeting tags: "+n),s.contentWindow.postMessage(f+n,"*")}else e.cM("Received unhandled message")}getSlotElementIdByEvent(e){let t=this._getFrameByEvent(e),s=t.parentElement?.parentElement?.id;return s||""}_getFrameByEvent(e){return Array.from(document.getElementsByTagName("iframe")).filter(t=>t.contentWindow===e.source)[0]}classifyZoneIds(e){const t=e.map(o.Nj).filter(o.yb);return{eligible:t.filter(o.xb).filter(o.pn),ineligible:t.filter(o.xj)}}applyExtraMarginBottom(t){if(t&&t.slot&&!t.isEmpty&&(t.creativeId||t.lineItemId||!t.isEmpty)){var s=t.slot.getSlotElementId();if(s){var o=document.getElementById(s);if(o)if((0,n.eq)(s)){var a=o?.closest(".js-zone-container");a.style.marginBottom="24px",e.cM("Applied extra margin to the bottom of "+s)}else e.cM("Not applying extra margin to the bottom of "+s+": shouldHaveReportButton = false");else e.cM("Not applying extra margin to the bottom of "+s+": resolved invalid adUnit element")}else e.cM("Not applying extra margin to the bottom of element: invalid adUnitElementId")}else e.cM("Not applying extra margin to the bottom of element: invalid SlotRenderEndedEvent")}async load(s=(0,n.kG)()){const r=this.opt.tlb_position===t.Above?["dfp-mlb","dfp-smlb"]:["dfp-mlb","dfp-smlb","dfp-tlb"];if(!this.isGptReady())return e.cM("Initializing..."),this.initGpt(),void googletag.cmd.push(()=>this.load(s));this.opt.adReportEnabled?(e.cM("Ad reporting enabled"),this.adReports=new a.t(googletag,this)):(e.cM("Ad reporting not enabled"),this.handleSlotRenderedNoAdReport()),this.opt.refresh?(e.cM("Ad refresh enabled"),this.adRefresh=new i(googletag,this.opt.refreshInterval)):e.cM("Ad refresh not enabled"),this.opt.brandSurveyEnabled&&(e.cM("Brand Survey enabled"),this.brandSurvey=new g(googletag,this.opt.brandSurveySettings)),e.cM("Attempting to load ads into ids: ",s);const{eligible:d,ineligible:l}=this.classifyZoneIds(s);if(this.initDebugPanel(googletag,d.concat(l)),d.forEach(e=>(0,o.cf)(e)),l.forEach(o.wo),0===d.length)return void e.cM("Found no ad ids on page");e.cM("Eligible ids:",d),this.opt.abd&&this.appendAdblockDetector();var c=googletag.pubads().getSlots();if(c){var p=c.filter(e=>s.indexOf(e.getSlotElementId())>=0);googletag.destroySlots(p)}this.opt.sf&&(googletag.pubads().setForceSafeFrame(!0),googletag.pubads().setSafeFrameConfig({allowOverlayExpansion:!0,allowPushExpansion:!0,sandbox:!0})),e.cM("Targeting consent: Checking...");let h=!1,m=!1;void 0!==this.opt.targeting_consent&&(m=!0,e.cM("Targeting consent: Parameter set"),e.cM("Targeting consent: Consent given? ",this.opt.targeting_consent),h=this.opt.targeting_consent),void 0!==this.opt.personalization_consent&&(e.cM("Personalization consent: Parameter set"),e.cM("Personalization consent: Consent given? ",this.opt.personalization_consent),h=h&&this.opt.personalization_consent),h=h&&m,this.setPrivacySettings(h),this.opt.ll||googletag.pubads().enableSingleRequest(),cam.sreEvent||(googletag.pubads().addEventListener("slotRenderEnded",e=>this.onSlotRendered(e)),cam.sreEvent=!0),await this.setTargeting();var u=d.filter(e=>!this.opt.ll||r.indexOf(e.id)<0),f=d.filter(e=>!!this.opt.ll&&r.indexOf(e.id)>=0);e.cM("Up front ids:",u),e.cM("Lazy loaded ids:",f),u.forEach(t=>{e.cM(`Defining ad for element ${t.id}`),this.defineSlot(t.id,googletag),t.setAttribute("data-dfp-zone","true")}),googletag.enableServices(),u.forEach(t=>{e.cM(`Displaying ad for element ${t.id}`),this.clc_options.dv_enabled?window.onDvtagReady(function(){googletag.display(t.id)}):googletag.cmd.push(()=>googletag.display(t.id))}),this.opt.ll&&(e.cM("Enabling lazy loading for GAM"),googletag.pubads().enableLazyLoad({fetchMarginPercent:0,renderMarginPercent:0}),e.cM("Setting up lazy loaded ad units"),f.forEach(t=>{e.cM(`Lazy loading - Defining Slot ${t.id}`),this.defineSlot(t.id,googletag)}),f.forEach(t=>{e.cM(`Lazy loading - Displaying ad for element ${t.id}`),this.clc_options.dv_enabled?window.onDvtagReady(function(){googletag.display(t.id)}):googletag.cmd.push(()=>googletag.display(t.id))}))}setPrivacySettings(e){e||googletag.pubads().setPrivacySettings({nonPersonalizedAds:!0})}async setTargeting(){if(!googletag)throw new Error("googletag not defined");let t=this.opt.targeting;if(!t)throw new Error("Targeting not defined (is "+typeof t+")");Object.keys(t).forEach(s=>{e.cM(`-> targeting - ${s}: ${t[s]}`),googletag.pubads().setTargeting(s,t[s])});let s=!1;if(void 0!==this.opt.targeting_consent&&(s=this.opt.targeting_consent),s){let t=(new c).getBrandSurveyEngagement();if(t.getTotalBrandImpressions().forEach((t,s)=>{e.cM(`-> targeting - BrandImpressions: ${s}: ${t}`),googletag.pubads().setTargeting("brand_"+s.toString()+"_impressions",t.toString())}),t.surveysIdsCompleted.forEach(t=>{e.cM(`-> targeting - SurveysTaken: ${t}`),googletag.pubads().setTargeting("survey_"+t+"_taken","true")}),this.clc_options.tgt_e&&this.getUserMetaPromise){let t=await this.getUserMetaPromise;t&&t.tgt_acct?(e.cM("-> targeting - User Account: "+t.tgt_acct),googletag.pubads().setTargeting("user-acct",t.tgt_acct.company_name),googletag.pubads().setTargeting("user_acct_top",t.tgt_acct.company_name),googletag.pubads().setTargeting("user_industry",t.tgt_acct.industry),googletag.pubads().setTargeting("user_employee_count",t.tgt_acct.employee_range)):e.cM("-> targeting - User Account: Not Found"),t&&Object.prototype.hasOwnProperty.call(t,"is_high_rep_earner")?(e.cM("-> targeting - High Rep Earner: "+t.is_high_rep_earner),googletag.pubads().setTargeting("IsHighRepEarner",t.is_high_rep_earner?"true":"false")):e.cM("-> targeting - High Rep Earner: not found")}if(localStorage){e.cM('Checking local storage for "jobs-last-clicked" key.');let t=localStorage.getItem("jobs-last-clicked")?"true":"false";e.cM(`-> targeting - jobs_clicked: ${t}`),googletag.pubads().setTargeting("jobs_clicked",t)}}}appendAdblockDetector(){const e=document.createElement("div");e.className="adsbox",e.id="clc-abd",e.style.position="absolute",e.style.pointerEvents="none",e.innerHTML=" ",document.body.appendChild(e)}onSlotRendered(s){try{const i=s.slot.getSlotElementId();let r=[];i||r.push("id=0");const d=document.getElementById(i);if(i&&!d&&r.push("el=0"),0!==r.length)return void this.stalled(r.join("&"));const{path:l,sizes:c,zone:g}=(0,n.Z7)(i,this.clc_options.upo);if(this.collapsed[g]&&s.isEmpty)return e.cM(`No line item for the element #${d.id}... collapsing.`),void(0,o.wo)(d);if(this.slotsRenderedEvents.push(s),s.lineItemId||s.creativeId||!s.isEmpty){e.cM(`Rendered ad for element #${d.id} [line item #${s.lineItemId}]`),e.cM(s);var a=d.parentElement;if(a.classList.contains("js-zone-container")){switch((0,o.cf)(a),i){case"dfp-tlb":this.opt.tlb_position===t.Above?a.classList.add("mb8"):a.classList.add("mt16");break;case"dfp-tag":a.classList.add("mb8");break;case"dfp-msb":a.classList.add("mt16");break;case"dfp-mlb":case"dfp-smlb":case"dfp-bmlb":a.classList.add("my8");break;case"dfp-isb":a.classList.add("mt24");break;case"dfp-m-aq":a.classList.add("my12"),a.classList.add("mx-auto")}(0,o.$Z)(a),(0,o.$Z)(d)}else e.cM(`No ad for element #${d.id}, collapsing`),e.cM(s),(0,o.wo)(d)}}catch(t){e.cM("Exception thrown onSlotRendered"),e.cM(t),this.stalled("e=1")}}stalled(e){(new Image).src=`https://${this.clc_options.h}/stalled.gif?${e}`}defineSlot(t,s){"dfp-isb"===t&&(e.cM("-> targeting - Sidebar: Inline"),s.pubads().setTargeting("Sidebar",["Inline"])),"dfp-tsb"===t&&(e.cM("-> targeting - Sidebar: Right"),s.pubads().setTargeting("Sidebar",["Right"]));const{path:o,sizes:a,zone:i}=(0,n.Z7)(t,this.clc_options.upo);e.cM(`Defining slot for ${t}: ${o}, sizes: ${JSON.stringify(a)}`),s.defineSlot(o,a,t).addService(s.pubads())}importGptLibrary(){this.gptImported||(this.gptImported=!0,void 0===this.opt.targeting_consent||this.opt.targeting_consent?(0,o.Gx)("https://securepubads.g.doubleclick.net/tag/js/gpt.js"):(0,o.Gx)("https://pagead2.googlesyndication.com/tag/js/gpt.js"))}importDvLibrary(){this.clc_options.dv_enabled&&(e.cM("Adding DoubleVerify library"),(0,o.Gx)("https://pub.doubleverify.com/dvtag/21569774/DV1289064/pub.js"),e.cM("Adding DoubleVerify onDvtagReady handler"),window.onDvtagReady=function(t,s=750){e.cM("DoubleVerify onDvtagReady called"),window.dvtag=window.dvtag||{},dvtag.cmd=dvtag.cmd||[];const n={callback:t,timeout:s,timestamp:(new Date).getTime()};dvtag.cmd.push(function(){dvtag.queueAdRequest(n)}),setTimeout(function(){const e=n.callback;n.callback=null,e&&e()},s)})}isGptReady(){return"undefined"!=typeof googletag&&!!googletag.apiReady}initGpt(){"undefined"==typeof googletag&&(window.googletag={cmd:(0,o.QZ)(()=>{this.importGptLibrary(),this.importDvLibrary()})})}getUserMeta(){if(this.opt.allowAccountTargetingForThisRequest&&this.clc_options.tgt_e&&this.clc_options.tgt_p>0){if(e.cM("Targeting enabled."),this.clc_options.tgt_p<100){e.cM("Targeting rate limit enabled. Rolling the dice...");const t=Math.floor(100*Math.random())+1;if(e.cM("Rolled "+t+" and the max is "+this.clc_options.tgt_p),t>this.clc_options.tgt_p)return void e.cM("Will not request targeting.")}return e.cM("Will request targeting."),function(e,t,s,n){if(t){const t=new Headers;return t.append("Accept","application/json"),async function(e,t={},s=5e3){if("number"!=typeof s&&null!=s&&!1!==s){if("string"!=typeof s)throw new Error("fetchWithTimeout: timeout must be a number");if(s=parseInt(s),isNaN(s))throw new Error("fetchWithTimeout: timeout must be a number (or string that can be parsed to a number)")}const n=new AbortController,{signal:o}=n,a=fetch(e,{...t,signal:o}),i=setTimeout(()=>n.abort(),s);try{const e=await a;return clearTimeout(i),e}catch(e){throw clearTimeout(i),e}}(s+"?"+new URLSearchParams({omni:e}),{method:"GET",mode:"cors",headers:t},n).then(e=>e.json())}return Promise.reject("No consent")}(this.opt.omni,this.opt.targeting_consent,this.clc_options.tgt_u,this.clc_options.tgt_to).catch(t=>{e.vU("Error fetching user account targeting"),e.vU(t)})}e.cM("Targeting disabled. Will not request account targeting data.")}initDebugPanel(t,s){e.cM("initDebugPanel"),e.cM("Not showing debug panel.")}},window.clcGamLoaderOptions&&(cam.init(),cam.load())})()})();</script> <footer id="footer" class="site-footer js-footer" role="contentinfo"> <div class="site-footer--container"> <nav class="site-footer--nav" aria-label="Footer"> <div class="site-footer--col"> <h5 class="-title"><a href="/">Mathematics</a></h5> <ul class="-list js-primary-footer-links"> <li><a class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 2 })" href="/tour">Tour</a></li> <li><a href="/help" class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 3 })">Help</a></li> <li><a class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 5 })" href="https://chat.stackexchange.com?tab=site&host=math.stackexchange.com">Chat</a></li> <li><a class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 13 })" href="/contact">Contact</a></li> <li><a class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 14 })" href="https://math.meta.stackexchange.com">Feedback</a></li> </ul> </div> <div class="site-footer--col"> <h5 class="-title"><a class="js-gps-track" data-gps-track="footer.click({ location: 2, link: 1 })" href="https://stackoverflow.co/">Company</a></h5> <ul class="-list"> <li><a href="https://stackoverflow.com" class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 15})">Stack Overflow</a></li> <li><a href="https://stackoverflow.co/teams/" class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 29 })">Teams</a></li> <li><a href="https://stackoverflow.co/advertising/" class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 21 })">Advertising</a></li> <li><a href="https://stackoverflow.co/advertising/employer-branding/" class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 20 })">Talent</a></li> <li><a class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 1 })" href="https://stackoverflow.co/">About</a></li> <li><a class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 27 })" href="https://stackoverflow.co/company/press/">Press</a></li> <li><a class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 7 })" href="https://stackoverflow.com/legal">Legal</a></li> <li><a class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 8 })" href="https://stackoverflow.com/legal/privacy-policy">Privacy Policy</a></li> <li><a class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 37 })" href="https://stackoverflow.com/legal/terms-of-service/public">Terms of Service</a></li> <li id="consent-footer-link"><button type="button" data-controller="cookie-settings" data-action="click->cookie-settings#toggle" class="s-btn s-btn__link py4 js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 38 })" data-consent-popup-loader="footer">Cookie Settings</button></li> <li><a class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link: 39 })" href="https://stackoverflow.com/legal/cookie-policy">Cookie Policy</a></li> </ul> </div> <div class="site-footer--col site-footer--categories-nav"> <div> <h5 class="-title"><a href="https://stackexchange.com" data-gps-track="footer.click({ location: 2, link: 30 })">Stack Exchange Network</a></h5> <ul class="-list"> <li> <a href="https://stackexchange.com/sites#technology" class="-link js-gps-track" data-gps-track="footer.click({ location: 2, link: 24 })"> Technology </a> </li> <li> <a href="https://stackexchange.com/sites#culturerecreation" class="-link js-gps-track" data-gps-track="footer.click({ location: 2, link: 24 })"> Culture & recreation </a> </li> <li> <a href="https://stackexchange.com/sites#lifearts" class="-link js-gps-track" data-gps-track="footer.click({ location: 2, link: 24 })"> Life & arts </a> </li> <li> <a href="https://stackexchange.com/sites#science" class="-link js-gps-track" data-gps-track="footer.click({ location: 2, link: 24 })"> Science </a> </li> <li> <a href="https://stackexchange.com/sites#professional" class="-link js-gps-track" data-gps-track="footer.click({ location: 2, link: 24 })"> Professional </a> </li> <li> <a href="https://stackexchange.com/sites#business" class="-link js-gps-track" data-gps-track="footer.click({ location: 2, link: 24 })"> Business </a> </li> <li class="mt16 md:mt0"> <a href="https://api.stackexchange.com/" class="-link js-gps-track" data-gps-track="footer.click({ location: 2, link: 24 })"> API </a> </li> <li> <a href="https://data.stackexchange.com/" class="-link js-gps-track" data-gps-track="footer.click({ location: 2, link: 24 })"> Data </a> </li> </ul> </div> </div> </nav> <div class="site-footer--copyright fs-fine md:mt24"> <ul class="-list -social md:mb8"> <li><a class="js-gps-track -link" data-gps-track="footer.click({ location: 2, link:4 })" href="https://stackoverflow.blog?blb=1">Blog</a></li> <li><a href="https://www.facebook.com/officialstackoverflow/" class="-link js-gps-track" data-gps-track="footer.click({ location: 2, link: 31 })">Facebook</a></li> <li><a href="https://twitter.com/stackoverflow" class="-link js-gps-track" data-gps-track="footer.click({ location: 2, link: 32 })">Twitter</a></li> <li><a href="https://linkedin.com/company/stack-overflow" class="-link js-gps-track" data-gps-track="footer.click({ location: 2, link: 33 })">LinkedIn</a></li> <li><a href="https://www.instagram.com/thestackoverflow" class="-link js-gps-track" data-gps-track="footer.click({ location: 2, link: 36 })">Instagram</a></li> </ul> <p class="md:mb0"> <span>Site design / logo © 2025 Stack Exchange Inc; </span> <span>user contributions licensed under </span> <a class="-link s-link td-underline" href="https://stackoverflow.com/help/licensing">CC BY-SA</a> <span>. </span> <span id="svnrev">rev 2025.2.25.23148</span> </p> </div> </div> </footer>  <script async src="https://www.googletagmanager.com/gtag/js?id=G-S812YQPLT2"></script> <script> window.dataLayer = window.dataLayer || []; function gtag() { dataLayer.push(arguments); } </script> <script> StackExchange.ready(function() { var ga3Settings = { autoLink: ["stackoverflow.blog","info.stackoverflowsolutions.com","stackoverflowsolutions.com"], sendTitles: true, tracker: window.ga, trackingCodes: [ 'UA-108242619-5' ], checkDimension: 'dimension42' }; var customGA4Dimensions = {}; customGA4Dimensions["requestid"] = "b3345b91-21be-4181-9c28-9927a7e52e11"; customGA4Dimensions["routename"] = "Questions/Show"; customGA4Dimensions["post_id"] = "1290529"; customGA4Dimensions["tags"] = "|real-analysis|"; var ga4Settings = { tracker: gtag, trackingCodes: [ 'G-S812YQPLT2' ], consentsToPerformanceCookies: "denied", consentsToTargetingCookies: "denied", eventParameters: customGA4Dimensions, checkForAdBlock: true, sendTitles: true, trackClicks: false, }; StackExchange.ga.init({ GA3: ga3Settings, GA4: ga4Settings }); StackExchange.ga.setDimension('dimension2', '|real-analysis|'); StackExchange.ga.setDimension('dimension3', 'Questions/Show'); StackExchange.ga.setDimension('dimension7', "1740539661.1914036275"); StackExchange.ga.trackPageView(); }); </script> <script src="https://cdn.cookielaw.org/scripttemplates/otSDKStub.js" charset="UTF-8" data-document-language="true" data-domain-script="cb0f3c87-b769-4e66-bbaa-377f9194216d"></script> <script defer src="https://cdn.sstatic.net/Js/modules/cookie-consent.en.js?v=36bebc18e04f"></script> <script>(function(){function c(){var b=a.contentDocument||a.contentWindow.document;if(b){var d=b.createElement('script');d.innerHTML="window.__CF$cv$params={r:'917cc3ae8824cdee',t:'MTc0MDUzOTY2MS4wMDAwMDA='};var a=document.createElement('script');a.nonce='';a.src='/cdn-cgi/challenge-platform/scripts/jsd/main.js';document.getElementsByTagName('head')[0].appendChild(a);";b.getElementsByTagName('head')[0].appendChild(d)}}if(document.body){var a=document.createElement('iframe');a.height=1;a.width=1;a.style.position='absolute';a.style.top=0;a.style.left=0;a.style.border='none';a.style.visibility='hidden';document.body.appendChild(a);if('loading'!==document.readyState)c();else if(window.addEventListener)document.addEventListener('DOMContentLoaded',c);else{var e=document.onreadystatechange||function(){};document.onreadystatechange=function(b){e(b);'loading'!==document.readyState&&(document.onreadystatechange=e,c())}}}})();</script></body> </html>

CINXE.COM

real analysis - Difference between boundary point & limit point. - Mathematics Stack Exchange