For a convergent sequence $limsup b_n = lim b_n$
$begingroup$
Happy new year!
I wish to prove the following result to practice with $limsup:
$
For a convergent sequence$ (b_n)$ we know $limsup b_n = lim b_n$
If $b_n$ is convergent, the set of accumulation points is definitely nonempty as it must contain the limit of $b_n$ and we know that $b_n$ is bounded. Therefore if $V$ is the set of accumulation points of $(b_n)$, we know that $limsup(b_n)=sup(V)$.
Intuitively and by previous exercises I know that since limits are unique, there must be precisely one accumulation point and therefore $V$ contains one element, namely $lim(b_n)$. We now conclude that:
$$limsup b_n = lim b_n $$
Was this argument convincing? Was I a bit sloppy, can I make this argument more precise, is it correct?
After this question I will attempt to prove a similar result for a divergent sequence $(b_n)to infty$ then $limsup b_n = infty$.
Edit/addition: $limsup b_n = infty$ follows immediately from unboundedness (from above) of a divergent sequence and the definition of $limsup$, which is nice.
real-analysis proof-verification limsup-and-liminf
asked Jan 2 at 15:33
Wesley StrikWesley Strik
2,123424
$endgroup$
add a comment |
$begingroup$
Happy new year!
I wish to prove the following result to practice with $limsup:
$
For a convergent sequence$ (b_n)$ we know $limsup b_n = lim b_n$
If $b_n$ is convergent, the set of accumulation points is definitely nonempty as it must contain the limit of $b_n$ and we know that $b_n$ is bounded. Therefore if $V$ is the set of accumulation points of $(b_n)$, we know that $limsup(b_n)=sup(V)$.
Intuitively and by previous exercises I know that since limits are unique, there must be precisely one accumulation point and therefore $V$ contains one element, namely $lim(b_n)$. We now conclude that:
$$limsup b_n = lim b_n $$
Was this argument convincing? Was I a bit sloppy, can I make this argument more precise, is it correct?
After this question I will attempt to prove a similar result for a divergent sequence $(b_n)to infty$ then $limsup b_n = infty$.
Edit/addition: $limsup b_n = infty$ follows immediately from unboundedness (from above) of a divergent sequence and the definition of $limsup$, which is nice.
real-analysis proof-verification limsup-and-liminf
asked Jan 2 at 15:33
Wesley StrikWesley Strik
2,123424
$endgroup$
$begingroup$
What is $sup b_n$?
$endgroup$
– El borito
Jan 2 at 15:41
1
$begingroup$
Oh, my bad, that's a typo
$endgroup$
– Wesley Strik
Jan 2 at 15:44
1
$begingroup$
Your argument is correct.
$endgroup$
– Song
Jan 2 at 17:21
add a comment |
$begingroup$
Happy new year!
I wish to prove the following result to practice with $limsup:
$
For a convergent sequence$ (b_n)$ we know $limsup b_n = lim b_n$
If $b_n$ is convergent, the set of accumulation points is definitely nonempty as it must contain the limit of $b_n$ and we know that $b_n$ is bounded. Therefore if $V$ is the set of accumulation points of $(b_n)$, we know that $limsup(b_n)=sup(V)$.
Intuitively and by previous exercises I know that since limits are unique, there must be precisely one accumulation point and therefore $V$ contains one element, namely $lim(b_n)$. We now conclude that:
$$limsup b_n = lim b_n $$
Was this argument convincing? Was I a bit sloppy, can I make this argument more precise, is it correct?
After this question I will attempt to prove a similar result for a divergent sequence $(b_n)to infty$ then $limsup b_n = infty$.
Edit/addition: $limsup b_n = infty$ follows immediately from unboundedness (from above) of a divergent sequence and the definition of $limsup$, which is nice.
real-analysis proof-verification limsup-and-liminf
asked Jan 2 at 15:33
Wesley StrikWesley Strik
2,123424
$endgroup$
Happy new year!
I wish to prove the following result to practice with $limsup:
$
For a convergent sequence$ (b_n)$ we know $limsup b_n = lim b_n$
If $b_n$ is convergent, the set of accumulation points is definitely nonempty as it must contain the limit of $b_n$ and we know that $b_n$ is bounded. Therefore if $V$ is the set of accumulation points of $(b_n)$, we know that $limsup(b_n)=sup(V)$.
Intuitively and by previous exercises I know that since limits are unique, there must be precisely one accumulation point and therefore $V$ contains one element, namely $lim(b_n)$. We now conclude that:
$$limsup b_n = lim b_n $$
Was this argument convincing? Was I a bit sloppy, can I make this argument more precise, is it correct?
After this question I will attempt to prove a similar result for a divergent sequence $(b_n)to infty$ then $limsup b_n = infty$.
Edit/addition: $limsup b_n = infty$ follows immediately from unboundedness (from above) of a divergent sequence and the definition of $limsup$, which is nice.
real-analysis proof-verification limsup-and-liminf
real-analysis proof-verification limsup-and-liminf
asked Jan 2 at 15:33
Wesley StrikWesley Strik
2,123424
asked Jan 2 at 15:33
Wesley StrikWesley Strik
2,123424
edited Jan 2 at 15:44
Wesley Strik
asked Jan 2 at 15:33
Wesley StrikWesley Strik
2,123424
asked Jan 2 at 15:33
Wesley StrikWesley Strik
2,123424
asked Jan 2 at 15:33
Wesley StrikWesley Strik
2,123424
2,123424
$begingroup$
What is $sup b_n$?
$endgroup$
– El borito
Jan 2 at 15:41
1
$begingroup$
Oh, my bad, that's a typo
$endgroup$
– Wesley Strik
Jan 2 at 15:44
1
$begingroup$
Your argument is correct.
$endgroup$
– Song
Jan 2 at 17:21
add a comment |
$begingroup$
What is $sup b_n$?
$endgroup$
– El borito
Jan 2 at 15:41
1
$begingroup$
Oh, my bad, that's a typo
$endgroup$
– Wesley Strik
Jan 2 at 15:44
1
$begingroup$
Your argument is correct.
$endgroup$
– Song
Jan 2 at 17:21
$begingroup$
What is $sup b_n$?
$endgroup$
– El borito
Jan 2 at 15:41
$begingroup$
What is $sup b_n$?
$endgroup$
– El borito
Jan 2 at 15:41
1
1
$begingroup$
Oh, my bad, that's a typo
$endgroup$
– Wesley Strik
Jan 2 at 15:44
$begingroup$
Oh, my bad, that's a typo
$endgroup$
– Wesley Strik
Jan 2 at 15:44
1
1
$begingroup$
Your argument is correct.
$endgroup$
– Song
Jan 2 at 17:21
$begingroup$
Your argument is correct.
$endgroup$
– Song
Jan 2 at 17:21
add a comment |
0
active
oldest
votes
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
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3059613%2ffor-a-convergent-sequence-limsup-b-n-lim-b-n%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3059613%2ffor-a-convergent-sequence-limsup-b-n-lim-b-n%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
$begingroup$
What is $sup b_n$?
$endgroup$
– El borito
Jan 2 at 15:41
1
$begingroup$
Oh, my bad, that's a typo
$endgroup$
– Wesley Strik
Jan 2 at 15:44
1
$begingroup$
Your argument is correct.
$endgroup$
– Song
Jan 2 at 17:21