Isolated points and limit points












0












$begingroup$


I’m reading a complex analysis textbook, and they do a brief review on real analysis. I’ve attached a screenshot of the page, and highlighted the statement in question. Here is the statement:
“Clearly, for the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$, each natural number is a limit point.”



My question is: since the sequence above is made up of only natural numbers, aren’t each of the numbers in the sequence isolated points, and hence, can’t be limit points? Not every neighbourhood of the point 12, say, contains infinitely many terms of the the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$. Am I missing something? enter image description here






real-analysis







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    There is a distinction between the limit points of a sequence $(f(n))_{nin Bbb N}$ and the limit points of the set ${f(n): nin Bbb N}$.
    $endgroup$
    – DanielWainfleet
    Dec 15 '18 at 6:01
















0












$begingroup$


I’m reading a complex analysis textbook, and they do a brief review on real analysis. I’ve attached a screenshot of the page, and highlighted the statement in question. Here is the statement:
“Clearly, for the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$, each natural number is a limit point.”



My question is: since the sequence above is made up of only natural numbers, aren’t each of the numbers in the sequence isolated points, and hence, can’t be limit points? Not every neighbourhood of the point 12, say, contains infinitely many terms of the the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$. Am I missing something? enter image description here






real-analysis







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    There is a distinction between the limit points of a sequence $(f(n))_{nin Bbb N}$ and the limit points of the set ${f(n): nin Bbb N}$.
    $endgroup$
    – DanielWainfleet
    Dec 15 '18 at 6:01














0












0








0





$begingroup$


I’m reading a complex analysis textbook, and they do a brief review on real analysis. I’ve attached a screenshot of the page, and highlighted the statement in question. Here is the statement:
“Clearly, for the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$, each natural number is a limit point.”



My question is: since the sequence above is made up of only natural numbers, aren’t each of the numbers in the sequence isolated points, and hence, can’t be limit points? Not every neighbourhood of the point 12, say, contains infinitely many terms of the the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$. Am I missing something? enter image description here






real-analysis







share|cite|improve this question









$endgroup$




I’m reading a complex analysis textbook, and they do a brief review on real analysis. I’ve attached a screenshot of the page, and highlighted the statement in question. Here is the statement:
“Clearly, for the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$, each natural number is a limit point.”



My question is: since the sequence above is made up of only natural numbers, aren’t each of the numbers in the sequence isolated points, and hence, can’t be limit points? Not every neighbourhood of the point 12, say, contains infinitely many terms of the the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$. Am I missing something? enter image description here






real-analysis




real-analysis






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 14 '18 at 20:39









Live Free or π HardLive Free or π Hard

481213




481213








  • 1




    $begingroup$
    There is a distinction between the limit points of a sequence $(f(n))_{nin Bbb N}$ and the limit points of the set ${f(n): nin Bbb N}$.
    $endgroup$
    – DanielWainfleet
    Dec 15 '18 at 6:01














  • 1




    $begingroup$
    There is a distinction between the limit points of a sequence $(f(n))_{nin Bbb N}$ and the limit points of the set ${f(n): nin Bbb N}$.
    $endgroup$
    – DanielWainfleet
    Dec 15 '18 at 6:01








1




1




$begingroup$
There is a distinction between the limit points of a sequence $(f(n))_{nin Bbb N}$ and the limit points of the set ${f(n): nin Bbb N}$.
$endgroup$
– DanielWainfleet
Dec 15 '18 at 6:01




$begingroup$
There is a distinction between the limit points of a sequence $(f(n))_{nin Bbb N}$ and the limit points of the set ${f(n): nin Bbb N}$.
$endgroup$
– DanielWainfleet
Dec 15 '18 at 6:01










1 Answer
1






active

oldest

votes


















1












$begingroup$

For every neighbourhood of $12$ there exist infinitely many indices where the sequence lies in this neighbourhood.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    @ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
    $endgroup$
    – Live Free or π Hard
    Dec 14 '18 at 21:02








  • 1




    $begingroup$
    Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
    $endgroup$
    – SmileyCraft
    Dec 14 '18 at 21:06












Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039853%2fisolated-points-and-limit-points%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

For every neighbourhood of $12$ there exist infinitely many indices where the sequence lies in this neighbourhood.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    @ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
    $endgroup$
    – Live Free or π Hard
    Dec 14 '18 at 21:02








  • 1




    $begingroup$
    Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
    $endgroup$
    – SmileyCraft
    Dec 14 '18 at 21:06
















1












$begingroup$

For every neighbourhood of $12$ there exist infinitely many indices where the sequence lies in this neighbourhood.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    @ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
    $endgroup$
    – Live Free or π Hard
    Dec 14 '18 at 21:02








  • 1




    $begingroup$
    Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
    $endgroup$
    – SmileyCraft
    Dec 14 '18 at 21:06














1












1








1





$begingroup$

For every neighbourhood of $12$ there exist infinitely many indices where the sequence lies in this neighbourhood.






share|cite|improve this answer









$endgroup$



For every neighbourhood of $12$ there exist infinitely many indices where the sequence lies in this neighbourhood.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 14 '18 at 20:43









SmileyCraftSmileyCraft

3,776519




3,776519












  • $begingroup$
    @ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
    $endgroup$
    – Live Free or π Hard
    Dec 14 '18 at 21:02








  • 1




    $begingroup$
    Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
    $endgroup$
    – SmileyCraft
    Dec 14 '18 at 21:06


















  • $begingroup$
    @ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
    $endgroup$
    – Live Free or π Hard
    Dec 14 '18 at 21:02








  • 1




    $begingroup$
    Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
    $endgroup$
    – SmileyCraft
    Dec 14 '18 at 21:06
















$begingroup$
@ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
$endgroup$
– Live Free or π Hard
Dec 14 '18 at 21:02






$begingroup$
@ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
$endgroup$
– Live Free or π Hard
Dec 14 '18 at 21:02






1




1




$begingroup$
Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
$endgroup$
– SmileyCraft
Dec 14 '18 at 21:06




$begingroup$
Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
$endgroup$
– SmileyCraft
Dec 14 '18 at 21:06


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039853%2fisolated-points-and-limit-points%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

How to change which sound is reproduced for terminal bell?

Image
3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question
Read more

Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

Image
up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
Read more

Can I use Tabulator js library in my java Spring + Thymeleaf project?

Image
0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"
Read more