x/n uniformly convergent?
$begingroup$
Let $f_n(x) : mathbb{R} rightarrow mathbb{R}$ be defined as $dfrac{x}{n}$. Is $f_n$ uniformly convergent?
With "$(a_n)$ is limited and $(b_n)$ converges to 0 $Rightarrow (a_nb_n)$ converges also to 0" i would guess that $xcdot dfrac{1}{n}$ converges uniformly but $x$ is not limited in this case. So does ist converge uniformly or not?
analysis uniform-convergence
$endgroup$
add a comment |
$begingroup$
Let $f_n(x) : mathbb{R} rightarrow mathbb{R}$ be defined as $dfrac{x}{n}$. Is $f_n$ uniformly convergent?
With "$(a_n)$ is limited and $(b_n)$ converges to 0 $Rightarrow (a_nb_n)$ converges also to 0" i would guess that $xcdot dfrac{1}{n}$ converges uniformly but $x$ is not limited in this case. So does ist converge uniformly or not?
analysis uniform-convergence
$endgroup$
$begingroup$
You need to specify an interval for the values of $x$.
$endgroup$
– GaC
Jan 30 '17 at 13:50
add a comment |
$begingroup$
Let $f_n(x) : mathbb{R} rightarrow mathbb{R}$ be defined as $dfrac{x}{n}$. Is $f_n$ uniformly convergent?
With "$(a_n)$ is limited and $(b_n)$ converges to 0 $Rightarrow (a_nb_n)$ converges also to 0" i would guess that $xcdot dfrac{1}{n}$ converges uniformly but $x$ is not limited in this case. So does ist converge uniformly or not?
analysis uniform-convergence
$endgroup$
Let $f_n(x) : mathbb{R} rightarrow mathbb{R}$ be defined as $dfrac{x}{n}$. Is $f_n$ uniformly convergent?
With "$(a_n)$ is limited and $(b_n)$ converges to 0 $Rightarrow (a_nb_n)$ converges also to 0" i would guess that $xcdot dfrac{1}{n}$ converges uniformly but $x$ is not limited in this case. So does ist converge uniformly or not?
analysis uniform-convergence
analysis uniform-convergence
edited Jan 30 '17 at 13:58
Arjihad
asked Jan 30 '17 at 13:43
ArjihadArjihad
390112
390112
$begingroup$
You need to specify an interval for the values of $x$.
$endgroup$
– GaC
Jan 30 '17 at 13:50
add a comment |
$begingroup$
You need to specify an interval for the values of $x$.
$endgroup$
– GaC
Jan 30 '17 at 13:50
$begingroup$
You need to specify an interval for the values of $x$.
$endgroup$
– GaC
Jan 30 '17 at 13:50
$begingroup$
You need to specify an interval for the values of $x$.
$endgroup$
– GaC
Jan 30 '17 at 13:50
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
It really depends on which space you are considering your set of functions: on every bounded set the convergence will be uniform, in every unbounded, not uniform.
$endgroup$
$begingroup$
Is $mathbb{R}$ limited?
$endgroup$
– Arjihad
Jan 30 '17 at 13:59
$begingroup$
of course not (if by limited you mean bounded)
$endgroup$
– b00n heT
Jan 30 '17 at 15:01
add a comment |
$begingroup$
If you are taking $x in [a, b]$ a fixed, closed sub-interval of $mathbb{R}$, then $|frac{x}{n}| leq |frac{b}{n}|$, which can be made arbitrarily small only depending on the value of $n$.
If you are taking $x in mathbb{R}$, then given any $epsilon > 0$, there exists $x in mathbb{R}$ such that $|frac{x}{n}| geq epsilon$ (it is enough to choose $x geq n epsilon$). So you cannot have $|frac{x}{n}| leq epsilon$ for all $x$ as soon as $n$ is big enough. Hence the convergence is not uniform on $mathbb{R}$.
$endgroup$
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%2f2120886%2fx-n-uniformly-convergent%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
$begingroup$
It really depends on which space you are considering your set of functions: on every bounded set the convergence will be uniform, in every unbounded, not uniform.
$endgroup$
$begingroup$
Is $mathbb{R}$ limited?
$endgroup$
– Arjihad
Jan 30 '17 at 13:59
$begingroup$
of course not (if by limited you mean bounded)
$endgroup$
– b00n heT
Jan 30 '17 at 15:01
add a comment |
$begingroup$
It really depends on which space you are considering your set of functions: on every bounded set the convergence will be uniform, in every unbounded, not uniform.
$endgroup$
$begingroup$
Is $mathbb{R}$ limited?
$endgroup$
– Arjihad
Jan 30 '17 at 13:59
$begingroup$
of course not (if by limited you mean bounded)
$endgroup$
– b00n heT
Jan 30 '17 at 15:01
add a comment |
$begingroup$
It really depends on which space you are considering your set of functions: on every bounded set the convergence will be uniform, in every unbounded, not uniform.
$endgroup$
It really depends on which space you are considering your set of functions: on every bounded set the convergence will be uniform, in every unbounded, not uniform.
answered Jan 30 '17 at 13:50
b00n heTb00n heT
10.4k12235
10.4k12235
$begingroup$
Is $mathbb{R}$ limited?
$endgroup$
– Arjihad
Jan 30 '17 at 13:59
$begingroup$
of course not (if by limited you mean bounded)
$endgroup$
– b00n heT
Jan 30 '17 at 15:01
add a comment |
$begingroup$
Is $mathbb{R}$ limited?
$endgroup$
– Arjihad
Jan 30 '17 at 13:59
$begingroup$
of course not (if by limited you mean bounded)
$endgroup$
– b00n heT
Jan 30 '17 at 15:01
$begingroup$
Is $mathbb{R}$ limited?
$endgroup$
– Arjihad
Jan 30 '17 at 13:59
$begingroup$
Is $mathbb{R}$ limited?
$endgroup$
– Arjihad
Jan 30 '17 at 13:59
$begingroup$
of course not (if by limited you mean bounded)
$endgroup$
– b00n heT
Jan 30 '17 at 15:01
$begingroup$
of course not (if by limited you mean bounded)
$endgroup$
– b00n heT
Jan 30 '17 at 15:01
add a comment |
$begingroup$
If you are taking $x in [a, b]$ a fixed, closed sub-interval of $mathbb{R}$, then $|frac{x}{n}| leq |frac{b}{n}|$, which can be made arbitrarily small only depending on the value of $n$.
If you are taking $x in mathbb{R}$, then given any $epsilon > 0$, there exists $x in mathbb{R}$ such that $|frac{x}{n}| geq epsilon$ (it is enough to choose $x geq n epsilon$). So you cannot have $|frac{x}{n}| leq epsilon$ for all $x$ as soon as $n$ is big enough. Hence the convergence is not uniform on $mathbb{R}$.
$endgroup$
add a comment |
$begingroup$
If you are taking $x in [a, b]$ a fixed, closed sub-interval of $mathbb{R}$, then $|frac{x}{n}| leq |frac{b}{n}|$, which can be made arbitrarily small only depending on the value of $n$.
If you are taking $x in mathbb{R}$, then given any $epsilon > 0$, there exists $x in mathbb{R}$ such that $|frac{x}{n}| geq epsilon$ (it is enough to choose $x geq n epsilon$). So you cannot have $|frac{x}{n}| leq epsilon$ for all $x$ as soon as $n$ is big enough. Hence the convergence is not uniform on $mathbb{R}$.
$endgroup$
add a comment |
$begingroup$
If you are taking $x in [a, b]$ a fixed, closed sub-interval of $mathbb{R}$, then $|frac{x}{n}| leq |frac{b}{n}|$, which can be made arbitrarily small only depending on the value of $n$.
If you are taking $x in mathbb{R}$, then given any $epsilon > 0$, there exists $x in mathbb{R}$ such that $|frac{x}{n}| geq epsilon$ (it is enough to choose $x geq n epsilon$). So you cannot have $|frac{x}{n}| leq epsilon$ for all $x$ as soon as $n$ is big enough. Hence the convergence is not uniform on $mathbb{R}$.
$endgroup$
If you are taking $x in [a, b]$ a fixed, closed sub-interval of $mathbb{R}$, then $|frac{x}{n}| leq |frac{b}{n}|$, which can be made arbitrarily small only depending on the value of $n$.
If you are taking $x in mathbb{R}$, then given any $epsilon > 0$, there exists $x in mathbb{R}$ such that $|frac{x}{n}| geq epsilon$ (it is enough to choose $x geq n epsilon$). So you cannot have $|frac{x}{n}| leq epsilon$ for all $x$ as soon as $n$ is big enough. Hence the convergence is not uniform on $mathbb{R}$.
answered Jan 30 '17 at 13:57
Nicolas PelletierNicolas Pelletier
18116
18116
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%2f2120886%2fx-n-uniformly-convergent%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 need to specify an interval for the values of $x$.
$endgroup$
– GaC
Jan 30 '17 at 13:50