$sup(lim) le lim(sup)$?
$begingroup$
$sup_{x,yin A} lim_{ntoinfty} f_n(x,y) le lim_{ntoinfty} sup_{x,yin A} f_n(x,y)$ ?
Proof) $f_nle sup f_n.\ lim f_nle lim(sup f_n)\ sup(lim f_n)le lim(sup f_n)$
I think that if I want this proof to be correct, I need some conditions.
(such as $lim f_n, lim(sup f_n)$ exists...)
Does $lim_{n to infty} sup_{x in X} f_n(x) = sup_{x in X} lim_{n to infty} f_n(x)$?
I read this question and all the counterexamples in the answers were not enough to dispute sup lim $le$ lim sup (Although they dispute sup lim = lim sup)
In summary, what I like to know is,
1.$sup(lim) le lim(sup)$?(with what condition?)
2.What are the required conditions for it to be correct?
Thank you for reading.
real-analysis
edited Jan 1 at 16:03
A.Γ.
22.8k32656
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
$endgroup$
add a comment |
$begingroup$
$sup_{x,yin A} lim_{ntoinfty} f_n(x,y) le lim_{ntoinfty} sup_{x,yin A} f_n(x,y)$ ?
Proof) $f_nle sup f_n.\ lim f_nle lim(sup f_n)\ sup(lim f_n)le lim(sup f_n)$
I think that if I want this proof to be correct, I need some conditions.
(such as $lim f_n, lim(sup f_n)$ exists...)
Does $lim_{n to infty} sup_{x in X} f_n(x) = sup_{x in X} lim_{n to infty} f_n(x)$?
I read this question and all the counterexamples in the answers were not enough to dispute sup lim $le$ lim sup (Although they dispute sup lim = lim sup)
In summary, what I like to know is,
1.$sup(lim) le lim(sup)$?(with what condition?)
2.What are the required conditions for it to be correct?
Thank you for reading.
real-analysis
edited Jan 1 at 16:03
A.Γ.
22.8k32656
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
$endgroup$
$begingroup$
You can use the backslash in front of commands like $limsup$limsupand $sin$ so they appear upright.
$endgroup$
– Chase Ryan Taylor
Oct 3 '17 at 5:19
$begingroup$
Look for a counterexample. Some situation where the inner sup on $x,y$ varies when $n$ varies.
$endgroup$
– GEdgar
Oct 3 '17 at 13:44
add a comment |
$begingroup$
$sup_{x,yin A} lim_{ntoinfty} f_n(x,y) le lim_{ntoinfty} sup_{x,yin A} f_n(x,y)$ ?
Proof) $f_nle sup f_n.\ lim f_nle lim(sup f_n)\ sup(lim f_n)le lim(sup f_n)$
I think that if I want this proof to be correct, I need some conditions.
(such as $lim f_n, lim(sup f_n)$ exists...)
Does $lim_{n to infty} sup_{x in X} f_n(x) = sup_{x in X} lim_{n to infty} f_n(x)$?
I read this question and all the counterexamples in the answers were not enough to dispute sup lim $le$ lim sup (Although they dispute sup lim = lim sup)
In summary, what I like to know is,
1.$sup(lim) le lim(sup)$?(with what condition?)
2.What are the required conditions for it to be correct?
Thank you for reading.
real-analysis
edited Jan 1 at 16:03
A.Γ.
22.8k32656
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
$endgroup$
$sup_{x,yin A} lim_{ntoinfty} f_n(x,y) le lim_{ntoinfty} sup_{x,yin A} f_n(x,y)$ ?
Proof) $f_nle sup f_n.\ lim f_nle lim(sup f_n)\ sup(lim f_n)le lim(sup f_n)$
I think that if I want this proof to be correct, I need some conditions.
(such as $lim f_n, lim(sup f_n)$ exists...)
Does $lim_{n to infty} sup_{x in X} f_n(x) = sup_{x in X} lim_{n to infty} f_n(x)$?
I read this question and all the counterexamples in the answers were not enough to dispute sup lim $le$ lim sup (Although they dispute sup lim = lim sup)
In summary, what I like to know is,
1.$sup(lim) le lim(sup)$?(with what condition?)
2.What are the required conditions for it to be correct?
Thank you for reading.
real-analysis
real-analysis
edited Jan 1 at 16:03
A.Γ.
22.8k32656
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
edited Jan 1 at 16:03
A.Γ.
22.8k32656
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
edited Jan 1 at 16:03
A.Γ.
22.8k32656
edited Jan 1 at 16:03
A.Γ.
22.8k32656
edited Jan 1 at 16:03
A.Γ.
22.8k32656
22.8k32656
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
1447
$begingroup$
You can use the backslash in front of commands like $limsup$limsupand $sin$ so they appear upright.
$endgroup$
– Chase Ryan Taylor
Oct 3 '17 at 5:19
$begingroup$
Look for a counterexample. Some situation where the inner sup on $x,y$ varies when $n$ varies.
$endgroup$
– GEdgar
Oct 3 '17 at 13:44
add a comment |
$begingroup$
You can use the backslash in front of commands like $limsup$limsupand $sin$ so they appear upright.
$endgroup$
– Chase Ryan Taylor
Oct 3 '17 at 5:19
$begingroup$
Look for a counterexample. Some situation where the inner sup on $x,y$ varies when $n$ varies.
$endgroup$
– GEdgar
Oct 3 '17 at 13:44
$begingroup$
You can use the backslash in front of commands like $limsup$
limsup and $sin$ so they appear upright.$endgroup$
– Chase Ryan Taylor
Oct 3 '17 at 5:19
$begingroup$
You can use the backslash in front of commands like $limsup$
limsup and $sin$ so they appear upright.$endgroup$
– Chase Ryan Taylor
Oct 3 '17 at 5:19
$begingroup$
Look for a counterexample. Some situation where the inner sup on $x,y$ varies when $n$ varies.
$endgroup$
– GEdgar
Oct 3 '17 at 13:44
$begingroup$
Look for a counterexample. Some situation where the inner sup on $x,y$ varies when $n$ varies.
$endgroup$
– GEdgar
Oct 3 '17 at 13:44
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%2f2455315%2fsup-lim-le-lim-sup%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%2f2455315%2fsup-lim-le-lim-sup%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
$begingroup$
You can use the backslash in front of commands like $limsup$
limsupand $sin$ so they appear upright.$endgroup$
– Chase Ryan Taylor
Oct 3 '17 at 5:19
$begingroup$
Look for a counterexample. Some situation where the inner sup on $x,y$ varies when $n$ varies.
$endgroup$
– GEdgar
Oct 3 '17 at 13:44