Uniformly convergent on each ccmpact set of $mathbb R$ but not on $mathbb R$
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As the title says, I am looking for a sequence of function which is uniformly convergent on all compact sets of $mathbb R$ but not on $mathbb R$.
I thought $f_n(x) = x/n$ is such a function since for any x in a bounded and closed subset of $mathbb R$. $sup(f_n(x)-f(x)) to 0$ as $ntoinfty$. But since $mathbb R$ is unbounded $x$ can get infinitely large thus the function sequence does not uniformly converge on $mathbb{R}$. I wanted to check if my understanding is correct. Thank you
real-analysis limits convergence uniform-convergence sequence-of-function
edited Jan 6 at 8:39
RRL
49.5k42573
asked Jan 6 at 8:24
Kaan YolseverKaan Yolsever
899
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add a comment |
$begingroup$
As the title says, I am looking for a sequence of function which is uniformly convergent on all compact sets of $mathbb R$ but not on $mathbb R$.
I thought $f_n(x) = x/n$ is such a function since for any x in a bounded and closed subset of $mathbb R$. $sup(f_n(x)-f(x)) to 0$ as $ntoinfty$. But since $mathbb R$ is unbounded $x$ can get infinitely large thus the function sequence does not uniformly converge on $mathbb{R}$. I wanted to check if my understanding is correct. Thank you
real-analysis limits convergence uniform-convergence sequence-of-function
edited Jan 6 at 8:39
RRL
49.5k42573
asked Jan 6 at 8:24
Kaan YolseverKaan Yolsever
899
$endgroup$
2
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Yes - good example.
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– RRL
Jan 6 at 8:38
1
$begingroup$
Check my edits to improve your MathJax skills.
$endgroup$
– RRL
Jan 6 at 8:41
1
$begingroup$
Here is another example of what you are looking for where it is a little more difficult to prove uniform convergence on the compact intervals.
$endgroup$
– RRL
Jan 6 at 8:52
$begingroup$
@RRL thanks a lot
$endgroup$
– Kaan Yolsever
Jan 6 at 8:59
add a comment |
$begingroup$
As the title says, I am looking for a sequence of function which is uniformly convergent on all compact sets of $mathbb R$ but not on $mathbb R$.
I thought $f_n(x) = x/n$ is such a function since for any x in a bounded and closed subset of $mathbb R$. $sup(f_n(x)-f(x)) to 0$ as $ntoinfty$. But since $mathbb R$ is unbounded $x$ can get infinitely large thus the function sequence does not uniformly converge on $mathbb{R}$. I wanted to check if my understanding is correct. Thank you
real-analysis limits convergence uniform-convergence sequence-of-function
edited Jan 6 at 8:39
RRL
49.5k42573
asked Jan 6 at 8:24
Kaan YolseverKaan Yolsever
899
$endgroup$
As the title says, I am looking for a sequence of function which is uniformly convergent on all compact sets of $mathbb R$ but not on $mathbb R$.
I thought $f_n(x) = x/n$ is such a function since for any x in a bounded and closed subset of $mathbb R$. $sup(f_n(x)-f(x)) to 0$ as $ntoinfty$. But since $mathbb R$ is unbounded $x$ can get infinitely large thus the function sequence does not uniformly converge on $mathbb{R}$. I wanted to check if my understanding is correct. Thank you
real-analysis limits convergence uniform-convergence sequence-of-function
real-analysis limits convergence uniform-convergence sequence-of-function
edited Jan 6 at 8:39
RRL
49.5k42573
asked Jan 6 at 8:24
Kaan YolseverKaan Yolsever
899
edited Jan 6 at 8:39
RRL
49.5k42573
asked Jan 6 at 8:24
Kaan YolseverKaan Yolsever
899
edited Jan 6 at 8:39
RRL
49.5k42573
edited Jan 6 at 8:39
RRL
49.5k42573
edited Jan 6 at 8:39
RRL
49.5k42573
49.5k42573
asked Jan 6 at 8:24
Kaan YolseverKaan Yolsever
899
asked Jan 6 at 8:24
Kaan YolseverKaan Yolsever
899
asked Jan 6 at 8:24
Kaan YolseverKaan Yolsever
899
899
2
$begingroup$
Yes - good example.
$endgroup$
– RRL
Jan 6 at 8:38
1
$begingroup$
Check my edits to improve your MathJax skills.
$endgroup$
– RRL
Jan 6 at 8:41
1
$begingroup$
Here is another example of what you are looking for where it is a little more difficult to prove uniform convergence on the compact intervals.
$endgroup$
– RRL
Jan 6 at 8:52
$begingroup$
@RRL thanks a lot
$endgroup$
– Kaan Yolsever
Jan 6 at 8:59
add a comment |
2
$begingroup$
Yes - good example.
$endgroup$
– RRL
Jan 6 at 8:38
1
$begingroup$
Check my edits to improve your MathJax skills.
$endgroup$
– RRL
Jan 6 at 8:41
1
$begingroup$
Here is another example of what you are looking for where it is a little more difficult to prove uniform convergence on the compact intervals.
$endgroup$
– RRL
Jan 6 at 8:52
$begingroup$
@RRL thanks a lot
$endgroup$
– Kaan Yolsever
Jan 6 at 8:59
2
2
$begingroup$
Yes - good example.
$endgroup$
– RRL
Jan 6 at 8:38
$begingroup$
Yes - good example.
$endgroup$
– RRL
Jan 6 at 8:38
1
1
$begingroup$
Check my edits to improve your MathJax skills.
$endgroup$
– RRL
Jan 6 at 8:41
$begingroup$
Check my edits to improve your MathJax skills.
$endgroup$
– RRL
Jan 6 at 8:41
1
1
$begingroup$
Here is another example of what you are looking for where it is a little more difficult to prove uniform convergence on the compact intervals.
$endgroup$
– RRL
Jan 6 at 8:52
$begingroup$
Here is another example of what you are looking for where it is a little more difficult to prove uniform convergence on the compact intervals.
$endgroup$
– RRL
Jan 6 at 8:52
$begingroup$
@RRL thanks a lot
$endgroup$
– Kaan Yolsever
Jan 6 at 8:59
$begingroup$
@RRL thanks a lot
$endgroup$
– Kaan Yolsever
Jan 6 at 8:59
add a comment |
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2
$begingroup$
Yes - good example.
$endgroup$
– RRL
Jan 6 at 8:38
1
$begingroup$
Check my edits to improve your MathJax skills.
$endgroup$
– RRL
Jan 6 at 8:41
1
$begingroup$
Here is another example of what you are looking for where it is a little more difficult to prove uniform convergence on the compact intervals.
$endgroup$
– RRL
Jan 6 at 8:52
$begingroup$
@RRL thanks a lot
$endgroup$
– Kaan Yolsever
Jan 6 at 8:59