Uniformly convergent on each ccmpact set of $mathbb R$ but not on $mathbb R$












3












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 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
















3












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 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














3












3








3


1



$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 6 at 8:39









RRL

49.5k42573




49.5k42573










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














  • 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










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