What is the main difference between pointwise and uniform convergence as defined here?
I have a little confusion here. I have seen the following several times and seem to be a bit confused as to differentiating them.
Let $E$ be a non-empty subset of $Bbb{R}$. A sequence of functions ${f_n}_{nin Bbb{N}},$ converges pointwise to $f$ on $E$ if and only if begin{align}f_n(x)to f(x),;forall,xin E.end{align}
On the other hand ${f_n}_{nin Bbb{N}},$ converges uniformly to $f$ on $E$ if and only if begin{align}f_n(x)to f(x),;forall,xin E.end{align}
QUESTION:
Why is $f_n(x)to f(x),;forall,xin E,$ is used for both uniform and pointwise convergence or I'm I missing something important? Can't we distinguish them?
real-analysis analysis definition uniform-convergence pointwise-convergence
asked 2 hours ago
Mike
1,498321
add a comment |
I have a little confusion here. I have seen the following several times and seem to be a bit confused as to differentiating them.
Let $E$ be a non-empty subset of $Bbb{R}$. A sequence of functions ${f_n}_{nin Bbb{N}},$ converges pointwise to $f$ on $E$ if and only if begin{align}f_n(x)to f(x),;forall,xin E.end{align}
On the other hand ${f_n}_{nin Bbb{N}},$ converges uniformly to $f$ on $E$ if and only if begin{align}f_n(x)to f(x),;forall,xin E.end{align}
QUESTION:
Why is $f_n(x)to f(x),;forall,xin E,$ is used for both uniform and pointwise convergence or I'm I missing something important? Can't we distinguish them?
real-analysis analysis definition uniform-convergence pointwise-convergence
asked 2 hours ago
Mike
1,498321
1
Please refer to the original definition, not the altered version. In your post, these are identical.
– xbh
1 hour ago
add a comment |
I have a little confusion here. I have seen the following several times and seem to be a bit confused as to differentiating them.
Let $E$ be a non-empty subset of $Bbb{R}$. A sequence of functions ${f_n}_{nin Bbb{N}},$ converges pointwise to $f$ on $E$ if and only if begin{align}f_n(x)to f(x),;forall,xin E.end{align}
On the other hand ${f_n}_{nin Bbb{N}},$ converges uniformly to $f$ on $E$ if and only if begin{align}f_n(x)to f(x),;forall,xin E.end{align}
QUESTION:
Why is $f_n(x)to f(x),;forall,xin E,$ is used for both uniform and pointwise convergence or I'm I missing something important? Can't we distinguish them?
real-analysis analysis definition uniform-convergence pointwise-convergence
asked 2 hours ago
Mike
1,498321
I have a little confusion here. I have seen the following several times and seem to be a bit confused as to differentiating them.
Let $E$ be a non-empty subset of $Bbb{R}$. A sequence of functions ${f_n}_{nin Bbb{N}},$ converges pointwise to $f$ on $E$ if and only if begin{align}f_n(x)to f(x),;forall,xin E.end{align}
On the other hand ${f_n}_{nin Bbb{N}},$ converges uniformly to $f$ on $E$ if and only if begin{align}f_n(x)to f(x),;forall,xin E.end{align}
QUESTION:
Why is $f_n(x)to f(x),;forall,xin E,$ is used for both uniform and pointwise convergence or I'm I missing something important? Can't we distinguish them?
real-analysis analysis definition uniform-convergence pointwise-convergence
real-analysis analysis definition uniform-convergence pointwise-convergence
asked 2 hours ago
Mike
1,498321
asked 2 hours ago
Mike
1,498321
edited 1 hour ago
asked 2 hours ago
Mike
1,498321
asked 2 hours ago
Mike
1,498321
asked 2 hours ago
Mike
1,498321
1,498321
1
Please refer to the original definition, not the altered version. In your post, these are identical.
– xbh
1 hour ago
add a comment |
1
Please refer to the original definition, not the altered version. In your post, these are identical.
– xbh
1 hour ago
1
1
Please refer to the original definition, not the altered version. In your post, these are identical.
– xbh
1 hour ago
Please refer to the original definition, not the altered version. In your post, these are identical.
– xbh
1 hour ago
add a comment |
3 Answers
3
active
oldest
votes
$f_n$ converges pointwise means for every $c>0$ for every $x$, there exists $N(x)$ such that $n>N(x)$ implies that $|f_n(x)-f(x)|<c$
$f_n$ converges uniformly means that for every $c>0$ there exists $N$ such that for every $x$, $n>N$ implies that $|f_n(x)-f(x)|<c$.
In the simply convergence, $N(x)$ depends of $x$ but for uniformly convergence one $N$ is chosen for every $x$.
answered 1 hour ago
Tsemo Aristide
56.2k11444
(+1) I already know this but want to know why $f_n(x)to f(x),;forall,xin E,$ is used for both uniform and pointwise convergence.
– Mike
1 hour ago
If $f_n$ converges uniformly, it convergence pointwise, so we can write $f_n(x)rightarrow f(x)$ for the both case. The difference is that for uniform convergence, if you draw the picture, you will see that the distance between the graph of $f_n$ and $f$ tends to $0$, this is not necessarily true for simply convergence.
– Tsemo Aristide
1 hour ago
That's so true.
– Mike
1 hour ago
Kindly check your post. Do you mean $f_n(n)=0$ or $f_n(x)=0$?
– Mike
1 hour ago
$f_n(n)=1, f_n(x)=0$ if $xneq n$
– Tsemo Aristide
1 hour ago
|
show 1 more comment
Uniform convergence is actually $mathcal L^infty$ convergence, i.e.
$$
f_n rightrightarrows f [x in E]!! iff !! sup_{x in E} vert f_n - fvert(x) to 0[n to infty].
$$
This is strictly stronger than pointwise convergence.
Alternatively, uniform convergence implies pointwise convergence, so $f_n to f$ in both cases.
answered 1 hour ago
xbh
5,7351522
1
Essential supremum and supremum coincide in the context of continuous functions. Pointwise convergence does imply uniform convergence in some particular context such as that of a sequence of $l$-Lipschitz functions where $l in mathbb R$.
– Matt A Pelto
18 mins ago
add a comment |
Sorry, but yes, you probably are missing something important, because the second statement in your post
On the other hand ${f_n}_{ninmathbb{N}}$, converges uniformly to $f$ on $E$ if and only if
$$f_n(x)to f(x),;forall,xin E.$$
is false. Without seeing your source, it's impossible to say what happened here, where this erroneous statement came from, and what exactly you're missing.
Are you sure the source says if and only if here? This certainly is NOT the definition of uniform convergence (unlike your first statement, which indeed is a definition of pointwise convergence, unless we want to expand it further in $varepsilon/delta$-language). This could be a theorem that states that uniform convergence implies pointwise convergence, which is a true theorem, but ONLY in this direction, so it cannot say if and only if.
Your best bet is to check the original source to find out what exactly it says there.
answered 4 mins ago
zipirovich
11.1k11631
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%2f3063473%2fwhat-is-the-main-difference-between-pointwise-and-uniform-convergence-as-defined%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$f_n$ converges pointwise means for every $c>0$ for every $x$, there exists $N(x)$ such that $n>N(x)$ implies that $|f_n(x)-f(x)|<c$
$f_n$ converges uniformly means that for every $c>0$ there exists $N$ such that for every $x$, $n>N$ implies that $|f_n(x)-f(x)|<c$.
In the simply convergence, $N(x)$ depends of $x$ but for uniformly convergence one $N$ is chosen for every $x$.
answered 1 hour ago
Tsemo Aristide
56.2k11444
(+1) I already know this but want to know why $f_n(x)to f(x),;forall,xin E,$ is used for both uniform and pointwise convergence.
– Mike
1 hour ago
If $f_n$ converges uniformly, it convergence pointwise, so we can write $f_n(x)rightarrow f(x)$ for the both case. The difference is that for uniform convergence, if you draw the picture, you will see that the distance between the graph of $f_n$ and $f$ tends to $0$, this is not necessarily true for simply convergence.
– Tsemo Aristide
1 hour ago
That's so true.
– Mike
1 hour ago
Kindly check your post. Do you mean $f_n(n)=0$ or $f_n(x)=0$?
– Mike
1 hour ago
$f_n(n)=1, f_n(x)=0$ if $xneq n$
– Tsemo Aristide
1 hour ago
|
show 1 more comment
$f_n$ converges pointwise means for every $c>0$ for every $x$, there exists $N(x)$ such that $n>N(x)$ implies that $|f_n(x)-f(x)|<c$
$f_n$ converges uniformly means that for every $c>0$ there exists $N$ such that for every $x$, $n>N$ implies that $|f_n(x)-f(x)|<c$.
In the simply convergence, $N(x)$ depends of $x$ but for uniformly convergence one $N$ is chosen for every $x$.
answered 1 hour ago
Tsemo Aristide
56.2k11444
(+1) I already know this but want to know why $f_n(x)to f(x),;forall,xin E,$ is used for both uniform and pointwise convergence.
– Mike
1 hour ago
If $f_n$ converges uniformly, it convergence pointwise, so we can write $f_n(x)rightarrow f(x)$ for the both case. The difference is that for uniform convergence, if you draw the picture, you will see that the distance between the graph of $f_n$ and $f$ tends to $0$, this is not necessarily true for simply convergence.
– Tsemo Aristide
1 hour ago
That's so true.
– Mike
1 hour ago
Kindly check your post. Do you mean $f_n(n)=0$ or $f_n(x)=0$?
– Mike
1 hour ago
$f_n(n)=1, f_n(x)=0$ if $xneq n$
– Tsemo Aristide
1 hour ago
|
show 1 more comment
$f_n$ converges pointwise means for every $c>0$ for every $x$, there exists $N(x)$ such that $n>N(x)$ implies that $|f_n(x)-f(x)|<c$
$f_n$ converges uniformly means that for every $c>0$ there exists $N$ such that for every $x$, $n>N$ implies that $|f_n(x)-f(x)|<c$.
In the simply convergence, $N(x)$ depends of $x$ but for uniformly convergence one $N$ is chosen for every $x$.
answered 1 hour ago
Tsemo Aristide
56.2k11444
$f_n$ converges pointwise means for every $c>0$ for every $x$, there exists $N(x)$ such that $n>N(x)$ implies that $|f_n(x)-f(x)|<c$
$f_n$ converges uniformly means that for every $c>0$ there exists $N$ such that for every $x$, $n>N$ implies that $|f_n(x)-f(x)|<c$.
In the simply convergence, $N(x)$ depends of $x$ but for uniformly convergence one $N$ is chosen for every $x$.
answered 1 hour ago
Tsemo Aristide
56.2k11444
answered 1 hour ago
Tsemo Aristide
56.2k11444
answered 1 hour ago
Tsemo Aristide
56.2k11444
answered 1 hour ago
Tsemo Aristide
56.2k11444
56.2k11444
(+1) I already know this but want to know why $f_n(x)to f(x),;forall,xin E,$ is used for both uniform and pointwise convergence.
– Mike
1 hour ago
If $f_n$ converges uniformly, it convergence pointwise, so we can write $f_n(x)rightarrow f(x)$ for the both case. The difference is that for uniform convergence, if you draw the picture, you will see that the distance between the graph of $f_n$ and $f$ tends to $0$, this is not necessarily true for simply convergence.
– Tsemo Aristide
1 hour ago
That's so true.
– Mike
1 hour ago
Kindly check your post. Do you mean $f_n(n)=0$ or $f_n(x)=0$?
– Mike
1 hour ago
$f_n(n)=1, f_n(x)=0$ if $xneq n$
– Tsemo Aristide
1 hour ago
|
show 1 more comment
(+1) I already know this but want to know why $f_n(x)to f(x),;forall,xin E,$ is used for both uniform and pointwise convergence.
– Mike
1 hour ago
If $f_n$ converges uniformly, it convergence pointwise, so we can write $f_n(x)rightarrow f(x)$ for the both case. The difference is that for uniform convergence, if you draw the picture, you will see that the distance between the graph of $f_n$ and $f$ tends to $0$, this is not necessarily true for simply convergence.
– Tsemo Aristide
1 hour ago
That's so true.
– Mike
1 hour ago
Kindly check your post. Do you mean $f_n(n)=0$ or $f_n(x)=0$?
– Mike
1 hour ago
$f_n(n)=1, f_n(x)=0$ if $xneq n$
– Tsemo Aristide
1 hour ago
(+1) I already know this but want to know why $f_n(x)to f(x),;forall,xin E,$ is used for both uniform and pointwise convergence.
– Mike
1 hour ago
(+1) I already know this but want to know why $f_n(x)to f(x),;forall,xin E,$ is used for both uniform and pointwise convergence.
– Mike
1 hour ago
If $f_n$ converges uniformly, it convergence pointwise, so we can write $f_n(x)rightarrow f(x)$ for the both case. The difference is that for uniform convergence, if you draw the picture, you will see that the distance between the graph of $f_n$ and $f$ tends to $0$, this is not necessarily true for simply convergence.
– Tsemo Aristide
1 hour ago
If $f_n$ converges uniformly, it convergence pointwise, so we can write $f_n(x)rightarrow f(x)$ for the both case. The difference is that for uniform convergence, if you draw the picture, you will see that the distance between the graph of $f_n$ and $f$ tends to $0$, this is not necessarily true for simply convergence.
– Tsemo Aristide
1 hour ago
That's so true.
– Mike
1 hour ago
That's so true.
– Mike
1 hour ago
Kindly check your post. Do you mean $f_n(n)=0$ or $f_n(x)=0$?
– Mike
1 hour ago
Kindly check your post. Do you mean $f_n(n)=0$ or $f_n(x)=0$?
– Mike
1 hour ago
$f_n(n)=1, f_n(x)=0$ if $xneq n$
– Tsemo Aristide
1 hour ago
$f_n(n)=1, f_n(x)=0$ if $xneq n$
– Tsemo Aristide
1 hour ago
|
show 1 more comment
Uniform convergence is actually $mathcal L^infty$ convergence, i.e.
$$
f_n rightrightarrows f [x in E]!! iff !! sup_{x in E} vert f_n - fvert(x) to 0[n to infty].
$$
This is strictly stronger than pointwise convergence.
Alternatively, uniform convergence implies pointwise convergence, so $f_n to f$ in both cases.
answered 1 hour ago
xbh
5,7351522
1
Essential supremum and supremum coincide in the context of continuous functions. Pointwise convergence does imply uniform convergence in some particular context such as that of a sequence of $l$-Lipschitz functions where $l in mathbb R$.
– Matt A Pelto
18 mins ago
add a comment |
Uniform convergence is actually $mathcal L^infty$ convergence, i.e.
$$
f_n rightrightarrows f [x in E]!! iff !! sup_{x in E} vert f_n - fvert(x) to 0[n to infty].
$$
This is strictly stronger than pointwise convergence.
Alternatively, uniform convergence implies pointwise convergence, so $f_n to f$ in both cases.
answered 1 hour ago
xbh
5,7351522
1
Essential supremum and supremum coincide in the context of continuous functions. Pointwise convergence does imply uniform convergence in some particular context such as that of a sequence of $l$-Lipschitz functions where $l in mathbb R$.
– Matt A Pelto
18 mins ago
add a comment |
Uniform convergence is actually $mathcal L^infty$ convergence, i.e.
$$
f_n rightrightarrows f [x in E]!! iff !! sup_{x in E} vert f_n - fvert(x) to 0[n to infty].
$$
This is strictly stronger than pointwise convergence.
Alternatively, uniform convergence implies pointwise convergence, so $f_n to f$ in both cases.
answered 1 hour ago
xbh
5,7351522
Uniform convergence is actually $mathcal L^infty$ convergence, i.e.
$$
f_n rightrightarrows f [x in E]!! iff !! sup_{x in E} vert f_n - fvert(x) to 0[n to infty].
$$
This is strictly stronger than pointwise convergence.
Alternatively, uniform convergence implies pointwise convergence, so $f_n to f$ in both cases.
answered 1 hour ago
xbh
5,7351522
answered 1 hour ago
xbh
5,7351522
answered 1 hour ago
xbh
5,7351522
answered 1 hour ago
xbh
5,7351522
5,7351522
1
Essential supremum and supremum coincide in the context of continuous functions. Pointwise convergence does imply uniform convergence in some particular context such as that of a sequence of $l$-Lipschitz functions where $l in mathbb R$.
– Matt A Pelto
18 mins ago
add a comment |
1
Essential supremum and supremum coincide in the context of continuous functions. Pointwise convergence does imply uniform convergence in some particular context such as that of a sequence of $l$-Lipschitz functions where $l in mathbb R$.
– Matt A Pelto
18 mins ago
1
1
Essential supremum and supremum coincide in the context of continuous functions. Pointwise convergence does imply uniform convergence in some particular context such as that of a sequence of $l$-Lipschitz functions where $l in mathbb R$.
– Matt A Pelto
18 mins ago
Essential supremum and supremum coincide in the context of continuous functions. Pointwise convergence does imply uniform convergence in some particular context such as that of a sequence of $l$-Lipschitz functions where $l in mathbb R$.
– Matt A Pelto
18 mins ago
add a comment |
Sorry, but yes, you probably are missing something important, because the second statement in your post
On the other hand ${f_n}_{ninmathbb{N}}$, converges uniformly to $f$ on $E$ if and only if
$$f_n(x)to f(x),;forall,xin E.$$
is false. Without seeing your source, it's impossible to say what happened here, where this erroneous statement came from, and what exactly you're missing.
Are you sure the source says if and only if here? This certainly is NOT the definition of uniform convergence (unlike your first statement, which indeed is a definition of pointwise convergence, unless we want to expand it further in $varepsilon/delta$-language). This could be a theorem that states that uniform convergence implies pointwise convergence, which is a true theorem, but ONLY in this direction, so it cannot say if and only if.
Your best bet is to check the original source to find out what exactly it says there.
answered 4 mins ago
zipirovich
11.1k11631
add a comment |
Sorry, but yes, you probably are missing something important, because the second statement in your post
On the other hand ${f_n}_{ninmathbb{N}}$, converges uniformly to $f$ on $E$ if and only if
$$f_n(x)to f(x),;forall,xin E.$$
is false. Without seeing your source, it's impossible to say what happened here, where this erroneous statement came from, and what exactly you're missing.
Are you sure the source says if and only if here? This certainly is NOT the definition of uniform convergence (unlike your first statement, which indeed is a definition of pointwise convergence, unless we want to expand it further in $varepsilon/delta$-language). This could be a theorem that states that uniform convergence implies pointwise convergence, which is a true theorem, but ONLY in this direction, so it cannot say if and only if.
Your best bet is to check the original source to find out what exactly it says there.
answered 4 mins ago
zipirovich
11.1k11631
add a comment |
Sorry, but yes, you probably are missing something important, because the second statement in your post
On the other hand ${f_n}_{ninmathbb{N}}$, converges uniformly to $f$ on $E$ if and only if
$$f_n(x)to f(x),;forall,xin E.$$
is false. Without seeing your source, it's impossible to say what happened here, where this erroneous statement came from, and what exactly you're missing.
Are you sure the source says if and only if here? This certainly is NOT the definition of uniform convergence (unlike your first statement, which indeed is a definition of pointwise convergence, unless we want to expand it further in $varepsilon/delta$-language). This could be a theorem that states that uniform convergence implies pointwise convergence, which is a true theorem, but ONLY in this direction, so it cannot say if and only if.
Your best bet is to check the original source to find out what exactly it says there.
answered 4 mins ago
zipirovich
11.1k11631
Sorry, but yes, you probably are missing something important, because the second statement in your post
On the other hand ${f_n}_{ninmathbb{N}}$, converges uniformly to $f$ on $E$ if and only if
$$f_n(x)to f(x),;forall,xin E.$$
is false. Without seeing your source, it's impossible to say what happened here, where this erroneous statement came from, and what exactly you're missing.
Are you sure the source says if and only if here? This certainly is NOT the definition of uniform convergence (unlike your first statement, which indeed is a definition of pointwise convergence, unless we want to expand it further in $varepsilon/delta$-language). This could be a theorem that states that uniform convergence implies pointwise convergence, which is a true theorem, but ONLY in this direction, so it cannot say if and only if.
Your best bet is to check the original source to find out what exactly it says there.
answered 4 mins ago
zipirovich
11.1k11631
answered 4 mins ago
zipirovich
11.1k11631
answered 4 mins ago
zipirovich
11.1k11631
answered 4 mins ago
zipirovich
11.1k11631
11.1k11631
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3063473%2fwhat-is-the-main-difference-between-pointwise-and-uniform-convergence-as-defined%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
Please refer to the original definition, not the altered version. In your post, these are identical.
– xbh
1 hour ago