Does $f_n$ converge uniformly? The 2019 Stack Overflow Developer Survey Results Are InDoes $(f_n)$ converge pointwise/uniformly on $I$?If $(f_n)_ngeq 1 to f$ uniformly and $(g_n)_ngeq 1 to g$ uniformly then $(f_n circ g_n)_ngeq 1 to fcirc g$ uniformlyWhy does the sequence of functions $f_n(x)=nx^n(1-x)$ not converge uniformly?Does $f_n(x)=nxe^-nxsin(x)$ converges uniformly on $mathbbR^+$?$(f_n) $ c.uniformly to $f$, $(g_n)$ c. uniformly $g$, $(f_n)(g_n) $ doesn't c. uniformly to $fg$. Informed duplicate.Does $f_n$ converge uniformly on $(0,1) $ and $(0,infty)?$Which of the following sequences $(f_n)$ converge uniformly on [0,1]?Does $f_n$ uniformly converge to $f$?How to show that a series of functions does not converge uniformly?Give a counter example: $f_ncdot g_n$ uniformly convergent then $f_n$ or $g_n$ uniformly convergent.
Did Section 31 appear in Star Trek: The Next Generation?
Protecting Dualbooting Windows from dangerous code (like rm -rf)
Feature engineering suggestion required
Why hard-Brexiteers don't insist on a hard border to prevent illegal immigration after Brexit?
Are there any other methods to apply to solving simultaneous equations?
Output the Arecibo Message
Aging parents with no investments
What is the closest word meaning "respect for time / mindful"
What to do when moving next to a bird sanctuary with a loosely-domesticated cat?
Worn-tile Scrabble
What does Linus Torvalds mean when he says that Git "never ever" tracks a file?
Who coined the term "madman theory"?
Apparent duplicates between Haynes service instructions and MOT
Why do UK politicians seemingly ignore opinion polls on Brexit?
If I score a critical hit on an 18 or higher, what are my chances of getting a critical hit if I roll 3d20?
How to save as into a customized destination on macOS?
What do the Banks children have against barley water?
What is the most effective way of iterating a std::vector and why?
How are circuits which use complex ICs normally simulated?
How come people say “Would of”?
Multiply Two Integer Polynomials
Why isn't the circumferential light around the M87 black hole's event horizon symmetric?
Did 3000BC Egyptians use meteoric iron weapons?
A poker game description that does not feel gimmicky
Does $f_n$ converge uniformly?
The 2019 Stack Overflow Developer Survey Results Are InDoes $(f_n)$ converge pointwise/uniformly on $I$?If $(f_n)_ngeq 1 to f$ uniformly and $(g_n)_ngeq 1 to g$ uniformly then $(f_n circ g_n)_ngeq 1 to fcirc g$ uniformlyWhy does the sequence of functions $f_n(x)=nx^n(1-x)$ not converge uniformly?Does $f_n(x)=nxe^-nxsin(x)$ converges uniformly on $mathbbR^+$?$(f_n) $ c.uniformly to $f$, $(g_n)$ c. uniformly $g$, $(f_n)(g_n) $ doesn't c. uniformly to $fg$. Informed duplicate.Does $f_n$ converge uniformly on $(0,1) $ and $(0,infty)?$Which of the following sequences $(f_n)$ converge uniformly on [0,1]?Does $f_n$ uniformly converge to $f$?How to show that a series of functions does not converge uniformly?Give a counter example: $f_ncdot g_n$ uniformly convergent then $f_n$ or $g_n$ uniformly convergent.
$begingroup$
So in my book I read two examples that the sequence of functions converges pointwise but not is an uniformly convergence. These sequences are:
$f_n : [0,1) rightarrow [0,1)$ given by $f_n (x) = x^n ;;forall n in mathbbN$ and $g_n : [0,1) rightarrow [0,1)$ given by $g_n (x) = fracxn ;;forall n in mathbbN$.
So I could understand that both sequences above has a convergence pointwise. I know why $(f_n)$ is not uniformly convergence but I don't understand why $(g_n)$ is not uniformly convergence!
For me, $forall epsilon >0$, there is $n_0 = n_0(epsilon) in mathbbN$ such that $d_infty(g_n, 0) = sup_x in [0,1]|g_n(x)-0(x)| =
sup_x in [0,1]|g_n(x)| =
sup_x in [0,1]Big| fracxnBig| =
fracxn < frac1n < epsilon.
$
For this, just take $n_0 > frac1epsilon$. Thus, the sequence $(g_n)$ is uniformly convergente to function $g equiv 0.$ But my book said that this sequence is not. Can you help me please?
real-analysis analysis uniform-convergence
edited Mar 30 at 18:16
José Carlos Santos
174k23133242
asked Mar 30 at 17:59
Thiago AlexandreThiago Alexandre
1347
$endgroup$
add a comment |
$begingroup$
So in my book I read two examples that the sequence of functions converges pointwise but not is an uniformly convergence. These sequences are:
$f_n : [0,1) rightarrow [0,1)$ given by $f_n (x) = x^n ;;forall n in mathbbN$ and $g_n : [0,1) rightarrow [0,1)$ given by $g_n (x) = fracxn ;;forall n in mathbbN$.
So I could understand that both sequences above has a convergence pointwise. I know why $(f_n)$ is not uniformly convergence but I don't understand why $(g_n)$ is not uniformly convergence!
For me, $forall epsilon >0$, there is $n_0 = n_0(epsilon) in mathbbN$ such that $d_infty(g_n, 0) = sup_x in [0,1]|g_n(x)-0(x)| =
sup_x in [0,1]|g_n(x)| =
sup_x in [0,1]Big| fracxnBig| =
fracxn < frac1n < epsilon.
$
For this, just take $n_0 > frac1epsilon$. Thus, the sequence $(g_n)$ is uniformly convergente to function $g equiv 0.$ But my book said that this sequence is not. Can you help me please?
real-analysis analysis uniform-convergence
edited Mar 30 at 18:16
José Carlos Santos
174k23133242
asked Mar 30 at 17:59
Thiago AlexandreThiago Alexandre
1347
$endgroup$
1
$begingroup$
Are you sure $g_n$ is not defined as $g_ncolon mathbbRtomathbbR$?
$endgroup$
– Clement C.
Mar 30 at 18:03
$begingroup$
I think that you are correct, because the author want to say that if you take out the compactness of the hypothesis, the Dini's theorem is not valid.
$endgroup$
– Thiago Alexandre
Mar 30 at 18:12
$begingroup$
That is good evidence, yes.
$endgroup$
– Clement C.
Mar 30 at 18:21
add a comment |
$begingroup$
So in my book I read two examples that the sequence of functions converges pointwise but not is an uniformly convergence. These sequences are:
$f_n : [0,1) rightarrow [0,1)$ given by $f_n (x) = x^n ;;forall n in mathbbN$ and $g_n : [0,1) rightarrow [0,1)$ given by $g_n (x) = fracxn ;;forall n in mathbbN$.
So I could understand that both sequences above has a convergence pointwise. I know why $(f_n)$ is not uniformly convergence but I don't understand why $(g_n)$ is not uniformly convergence!
For me, $forall epsilon >0$, there is $n_0 = n_0(epsilon) in mathbbN$ such that $d_infty(g_n, 0) = sup_x in [0,1]|g_n(x)-0(x)| =
sup_x in [0,1]|g_n(x)| =
sup_x in [0,1]Big| fracxnBig| =
fracxn < frac1n < epsilon.
$
For this, just take $n_0 > frac1epsilon$. Thus, the sequence $(g_n)$ is uniformly convergente to function $g equiv 0.$ But my book said that this sequence is not. Can you help me please?
real-analysis analysis uniform-convergence
edited Mar 30 at 18:16
José Carlos Santos
174k23133242
asked Mar 30 at 17:59
Thiago AlexandreThiago Alexandre
1347
$endgroup$
So in my book I read two examples that the sequence of functions converges pointwise but not is an uniformly convergence. These sequences are:
$f_n : [0,1) rightarrow [0,1)$ given by $f_n (x) = x^n ;;forall n in mathbbN$ and $g_n : [0,1) rightarrow [0,1)$ given by $g_n (x) = fracxn ;;forall n in mathbbN$.
So I could understand that both sequences above has a convergence pointwise. I know why $(f_n)$ is not uniformly convergence but I don't understand why $(g_n)$ is not uniformly convergence!
For me, $forall epsilon >0$, there is $n_0 = n_0(epsilon) in mathbbN$ such that $d_infty(g_n, 0) = sup_x in [0,1]|g_n(x)-0(x)| =
sup_x in [0,1]|g_n(x)| =
sup_x in [0,1]Big| fracxnBig| =
fracxn < frac1n < epsilon.
$
For this, just take $n_0 > frac1epsilon$. Thus, the sequence $(g_n)$ is uniformly convergente to function $g equiv 0.$ But my book said that this sequence is not. Can you help me please?
real-analysis analysis uniform-convergence
real-analysis analysis uniform-convergence
edited Mar 30 at 18:16
José Carlos Santos
174k23133242
asked Mar 30 at 17:59
Thiago AlexandreThiago Alexandre
1347
edited Mar 30 at 18:16
José Carlos Santos
174k23133242
asked Mar 30 at 17:59
Thiago AlexandreThiago Alexandre
1347
edited Mar 30 at 18:16
José Carlos Santos
174k23133242
edited Mar 30 at 18:16
José Carlos Santos
174k23133242
edited Mar 30 at 18:16
José Carlos Santos
174k23133242
174k23133242
asked Mar 30 at 17:59
Thiago AlexandreThiago Alexandre
1347
asked Mar 30 at 17:59
Thiago AlexandreThiago Alexandre
1347
asked Mar 30 at 17:59
Thiago AlexandreThiago Alexandre
1347
1347
1
$begingroup$
Are you sure $g_n$ is not defined as $g_ncolon mathbbRtomathbbR$?
$endgroup$
– Clement C.
Mar 30 at 18:03
$begingroup$
I think that you are correct, because the author want to say that if you take out the compactness of the hypothesis, the Dini's theorem is not valid.
$endgroup$
– Thiago Alexandre
Mar 30 at 18:12
$begingroup$
That is good evidence, yes.
$endgroup$
– Clement C.
Mar 30 at 18:21
add a comment |
1
$begingroup$
Are you sure $g_n$ is not defined as $g_ncolon mathbbRtomathbbR$?
$endgroup$
– Clement C.
Mar 30 at 18:03
$begingroup$
I think that you are correct, because the author want to say that if you take out the compactness of the hypothesis, the Dini's theorem is not valid.
$endgroup$
– Thiago Alexandre
Mar 30 at 18:12
$begingroup$
That is good evidence, yes.
$endgroup$
– Clement C.
Mar 30 at 18:21
1
1
$begingroup$
Are you sure $g_n$ is not defined as $g_ncolon mathbbRtomathbbR$?
$endgroup$
– Clement C.
Mar 30 at 18:03
$begingroup$
Are you sure $g_n$ is not defined as $g_ncolon mathbbRtomathbbR$?
$endgroup$
– Clement C.
Mar 30 at 18:03
$begingroup$
I think that you are correct, because the author want to say that if you take out the compactness of the hypothesis, the Dini's theorem is not valid.
$endgroup$
– Thiago Alexandre
Mar 30 at 18:12
$begingroup$
I think that you are correct, because the author want to say that if you take out the compactness of the hypothesis, the Dini's theorem is not valid.
$endgroup$
– Thiago Alexandre
Mar 30 at 18:12
$begingroup$
That is good evidence, yes.
$endgroup$
– Clement C.
Mar 30 at 18:21
$begingroup$
That is good evidence, yes.
$endgroup$
– Clement C.
Mar 30 at 18:21
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Of course you don't understand why $(g_n)_ninmathbb N$ doesn't converge uniformly. It does! And your proof is fine.
However, it doesn't converge uniformly in unbounded intervals (such as $mathbb R$ or $[0,infty)$). Perhaps that that's what your textbook says.
answered Mar 30 at 18:03
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
$begingroup$
I think that you are correct, because the author want to say that if you take out the compactness of the hypothesis, the Dini's theorem is not valid.
$endgroup$
– Thiago Alexandre
Mar 30 at 18:12
$begingroup$
Indeed, it is not.
$endgroup$
– José Carlos Santos
Mar 30 at 18:14
add a comment |
$begingroup$
You're book may be concern with unbounded intervals. In such case $fracxn$ is not uniformly convergent.
The fuction is uniformly convergent if it uniformly approaches to its convergent point so the given sequence fails in to converge uniformly on unbounded intervals
answered Mar 30 at 18:25
WKhanWKhan
588
$endgroup$
add a comment |
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%2f3168585%2fdoes-f-n-converge-uniformly%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Of course you don't understand why $(g_n)_ninmathbb N$ doesn't converge uniformly. It does! And your proof is fine.
However, it doesn't converge uniformly in unbounded intervals (such as $mathbb R$ or $[0,infty)$). Perhaps that that's what your textbook says.
answered Mar 30 at 18:03
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
$begingroup$
I think that you are correct, because the author want to say that if you take out the compactness of the hypothesis, the Dini's theorem is not valid.
$endgroup$
– Thiago Alexandre
Mar 30 at 18:12
$begingroup$
Indeed, it is not.
$endgroup$
– José Carlos Santos
Mar 30 at 18:14
add a comment |
$begingroup$
Of course you don't understand why $(g_n)_ninmathbb N$ doesn't converge uniformly. It does! And your proof is fine.
However, it doesn't converge uniformly in unbounded intervals (such as $mathbb R$ or $[0,infty)$). Perhaps that that's what your textbook says.
answered Mar 30 at 18:03
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
$begingroup$
I think that you are correct, because the author want to say that if you take out the compactness of the hypothesis, the Dini's theorem is not valid.
$endgroup$
– Thiago Alexandre
Mar 30 at 18:12
$begingroup$
Indeed, it is not.
$endgroup$
– José Carlos Santos
Mar 30 at 18:14
add a comment |
$begingroup$
Of course you don't understand why $(g_n)_ninmathbb N$ doesn't converge uniformly. It does! And your proof is fine.
However, it doesn't converge uniformly in unbounded intervals (such as $mathbb R$ or $[0,infty)$). Perhaps that that's what your textbook says.
answered Mar 30 at 18:03
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
Of course you don't understand why $(g_n)_ninmathbb N$ doesn't converge uniformly. It does! And your proof is fine.
However, it doesn't converge uniformly in unbounded intervals (such as $mathbb R$ or $[0,infty)$). Perhaps that that's what your textbook says.
answered Mar 30 at 18:03
José Carlos SantosJosé Carlos Santos
174k23133242
answered Mar 30 at 18:03
José Carlos SantosJosé Carlos Santos
174k23133242
answered Mar 30 at 18:03
José Carlos SantosJosé Carlos Santos
174k23133242
answered Mar 30 at 18:03
José Carlos SantosJosé Carlos Santos
174k23133242
174k23133242
$begingroup$
I think that you are correct, because the author want to say that if you take out the compactness of the hypothesis, the Dini's theorem is not valid.
$endgroup$
– Thiago Alexandre
Mar 30 at 18:12
$begingroup$
Indeed, it is not.
$endgroup$
– José Carlos Santos
Mar 30 at 18:14
add a comment |
$begingroup$
I think that you are correct, because the author want to say that if you take out the compactness of the hypothesis, the Dini's theorem is not valid.
$endgroup$
– Thiago Alexandre
Mar 30 at 18:12
$begingroup$
Indeed, it is not.
$endgroup$
– José Carlos Santos
Mar 30 at 18:14
$begingroup$
I think that you are correct, because the author want to say that if you take out the compactness of the hypothesis, the Dini's theorem is not valid.
$endgroup$
– Thiago Alexandre
Mar 30 at 18:12
$begingroup$
I think that you are correct, because the author want to say that if you take out the compactness of the hypothesis, the Dini's theorem is not valid.
$endgroup$
– Thiago Alexandre
Mar 30 at 18:12
$begingroup$
Indeed, it is not.
$endgroup$
– José Carlos Santos
Mar 30 at 18:14
$begingroup$
Indeed, it is not.
$endgroup$
– José Carlos Santos
Mar 30 at 18:14
add a comment |
$begingroup$
You're book may be concern with unbounded intervals. In such case $fracxn$ is not uniformly convergent.
The fuction is uniformly convergent if it uniformly approaches to its convergent point so the given sequence fails in to converge uniformly on unbounded intervals
answered Mar 30 at 18:25
WKhanWKhan
588
$endgroup$
add a comment |
$begingroup$
You're book may be concern with unbounded intervals. In such case $fracxn$ is not uniformly convergent.
The fuction is uniformly convergent if it uniformly approaches to its convergent point so the given sequence fails in to converge uniformly on unbounded intervals
answered Mar 30 at 18:25
WKhanWKhan
588
$endgroup$
add a comment |
$begingroup$
You're book may be concern with unbounded intervals. In such case $fracxn$ is not uniformly convergent.
The fuction is uniformly convergent if it uniformly approaches to its convergent point so the given sequence fails in to converge uniformly on unbounded intervals
answered Mar 30 at 18:25
WKhanWKhan
588
$endgroup$
You're book may be concern with unbounded intervals. In such case $fracxn$ is not uniformly convergent.
The fuction is uniformly convergent if it uniformly approaches to its convergent point so the given sequence fails in to converge uniformly on unbounded intervals
answered Mar 30 at 18:25
WKhanWKhan
588
answered Mar 30 at 18:25
WKhanWKhan
588
answered Mar 30 at 18:25
WKhanWKhan
588
answered Mar 30 at 18:25
WKhanWKhan
588
588
add a comment |
add a comment |
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%2f3168585%2fdoes-f-n-converge-uniformly%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
1
$begingroup$
Are you sure $g_n$ is not defined as $g_ncolon mathbbRtomathbbR$?
$endgroup$
– Clement C.
Mar 30 at 18:03
$begingroup$
I think that you are correct, because the author want to say that if you take out the compactness of the hypothesis, the Dini's theorem is not valid.
$endgroup$
– Thiago Alexandre
Mar 30 at 18:12
$begingroup$
That is good evidence, yes.
$endgroup$
– Clement C.
Mar 30 at 18:21