Example of a uniformly convergent subsequence of a sequence of function which is pointwise convergent.












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let $ {f_n} $ be a sequence of function on any subset of $ mathbb R $ and it converges pointwise to zero. Is it possible to get a subsequence which converges uniformly to zero?
Thanks in advance.






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    0












    $begingroup$


    let $ {f_n} $ be a sequence of function on any subset of $ mathbb R $ and it converges pointwise to zero. Is it possible to get a subsequence which converges uniformly to zero?
    Thanks in advance.






    real-analysis







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      let $ {f_n} $ be a sequence of function on any subset of $ mathbb R $ and it converges pointwise to zero. Is it possible to get a subsequence which converges uniformly to zero?
      Thanks in advance.






      real-analysis







      share|cite|improve this question









      $endgroup$




      let $ {f_n} $ be a sequence of function on any subset of $ mathbb R $ and it converges pointwise to zero. Is it possible to get a subsequence which converges uniformly to zero?
      Thanks in advance.






      real-analysis




      real-analysis






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      asked Dec 31 '18 at 18:00









      suchanda adhikarisuchanda adhikari

      586




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          3 Answers
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          $begingroup$

          If $$f_n(x)=begin{cases}1,&(xge n),\0,&(x<n)end{cases}$$it's clear that $f_n(x)to0$ for every $x$ but no subsequence converges uniformly






          share|cite|improve this answer









          $endgroup$





















            0












            $begingroup$

            A classic example is the sequence $f_n(x)=x^n, n=1,2,dots $ on $(0,1).$






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              can anyone give me an easy example of a pointwise convergent sequence which has a uniformly convergent subsequenice?
              $endgroup$
              – suchanda adhikari
              Dec 31 '18 at 18:39










            • $begingroup$
              @suchandaadhikari That's easy. The sequence $0,0,0,dots$ will work.
              $endgroup$
              – zhw.
              Dec 31 '18 at 18:40










            • $begingroup$
              nice one ..can you give me an example of a sequence of function which is not uniformly convergent but pointwise convergent and has a uniformly convergent subsequence please?
              $endgroup$
              – suchanda adhikari
              Dec 31 '18 at 18:55






            • 1




              $begingroup$
              @suchandaadhikari $x,0,x^2,0,x^3,0,dots $
              $endgroup$
              – zhw.
              Dec 31 '18 at 19:03



















            0












            $begingroup$

            Consider the sequnce $(f_n)_{ninmathbb N}$ of functions from $mathbb R$ into itself defined by$$f_n(x)=begin{cases}dfrac1{(1+x^2)^n}&text{ if }xneq0\0&text{ otherwise.}end{cases}$$and prove that no subsequence of this sequence converges uniformly. However, $(f_n)_{ninmathbb N}$ converges pointwise to the null function.






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              so you mean there is no possibility to get a uniformly convergent subsequence from any sequence of function which converges pointwise to zero?
              $endgroup$
              – suchanda adhikari
              Dec 31 '18 at 18:23










            • $begingroup$
              Not at all. I only claimed that for the specific sequence that I defined.
              $endgroup$
              – José Carlos Santos
              Dec 31 '18 at 18:28










            • $begingroup$
              Correct, however there are theorems about getting uniform convergence outside a set of small measure.
              $endgroup$
              – Charlie Frohman
              Dec 31 '18 at 18:28










            • $begingroup$
              en.m.wikipedia.org/wiki/Egorov%27s_theorem
              $endgroup$
              – Charlie Frohman
              Dec 31 '18 at 18:29











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            3 Answers
            3






            active

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            3 Answers
            3






            active

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            active

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            1












            $begingroup$

            If $$f_n(x)=begin{cases}1,&(xge n),\0,&(x<n)end{cases}$$it's clear that $f_n(x)to0$ for every $x$ but no subsequence converges uniformly






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              If $$f_n(x)=begin{cases}1,&(xge n),\0,&(x<n)end{cases}$$it's clear that $f_n(x)to0$ for every $x$ but no subsequence converges uniformly






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                If $$f_n(x)=begin{cases}1,&(xge n),\0,&(x<n)end{cases}$$it's clear that $f_n(x)to0$ for every $x$ but no subsequence converges uniformly






                share|cite|improve this answer









                $endgroup$



                If $$f_n(x)=begin{cases}1,&(xge n),\0,&(x<n)end{cases}$$it's clear that $f_n(x)to0$ for every $x$ but no subsequence converges uniformly







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 31 '18 at 18:14









                David C. UllrichDavid C. Ullrich

                61k43994




                61k43994























                    0












                    $begingroup$

                    A classic example is the sequence $f_n(x)=x^n, n=1,2,dots $ on $(0,1).$






                    share|cite|improve this answer









                    $endgroup$













                    • $begingroup$
                      can anyone give me an easy example of a pointwise convergent sequence which has a uniformly convergent subsequenice?
                      $endgroup$
                      – suchanda adhikari
                      Dec 31 '18 at 18:39










                    • $begingroup$
                      @suchandaadhikari That's easy. The sequence $0,0,0,dots$ will work.
                      $endgroup$
                      – zhw.
                      Dec 31 '18 at 18:40










                    • $begingroup$
                      nice one ..can you give me an example of a sequence of function which is not uniformly convergent but pointwise convergent and has a uniformly convergent subsequence please?
                      $endgroup$
                      – suchanda adhikari
                      Dec 31 '18 at 18:55






                    • 1




                      $begingroup$
                      @suchandaadhikari $x,0,x^2,0,x^3,0,dots $
                      $endgroup$
                      – zhw.
                      Dec 31 '18 at 19:03
















                    0












                    $begingroup$

                    A classic example is the sequence $f_n(x)=x^n, n=1,2,dots $ on $(0,1).$






                    share|cite|improve this answer









                    $endgroup$













                    • $begingroup$
                      can anyone give me an easy example of a pointwise convergent sequence which has a uniformly convergent subsequenice?
                      $endgroup$
                      – suchanda adhikari
                      Dec 31 '18 at 18:39










                    • $begingroup$
                      @suchandaadhikari That's easy. The sequence $0,0,0,dots$ will work.
                      $endgroup$
                      – zhw.
                      Dec 31 '18 at 18:40










                    • $begingroup$
                      nice one ..can you give me an example of a sequence of function which is not uniformly convergent but pointwise convergent and has a uniformly convergent subsequence please?
                      $endgroup$
                      – suchanda adhikari
                      Dec 31 '18 at 18:55






                    • 1




                      $begingroup$
                      @suchandaadhikari $x,0,x^2,0,x^3,0,dots $
                      $endgroup$
                      – zhw.
                      Dec 31 '18 at 19:03














                    0












                    0








                    0





                    $begingroup$

                    A classic example is the sequence $f_n(x)=x^n, n=1,2,dots $ on $(0,1).$






                    share|cite|improve this answer









                    $endgroup$



                    A classic example is the sequence $f_n(x)=x^n, n=1,2,dots $ on $(0,1).$







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Dec 31 '18 at 18:23









                    zhw.zhw.

                    73.6k43175




                    73.6k43175












                    • $begingroup$
                      can anyone give me an easy example of a pointwise convergent sequence which has a uniformly convergent subsequenice?
                      $endgroup$
                      – suchanda adhikari
                      Dec 31 '18 at 18:39










                    • $begingroup$
                      @suchandaadhikari That's easy. The sequence $0,0,0,dots$ will work.
                      $endgroup$
                      – zhw.
                      Dec 31 '18 at 18:40










                    • $begingroup$
                      nice one ..can you give me an example of a sequence of function which is not uniformly convergent but pointwise convergent and has a uniformly convergent subsequence please?
                      $endgroup$
                      – suchanda adhikari
                      Dec 31 '18 at 18:55






                    • 1




                      $begingroup$
                      @suchandaadhikari $x,0,x^2,0,x^3,0,dots $
                      $endgroup$
                      – zhw.
                      Dec 31 '18 at 19:03


















                    • $begingroup$
                      can anyone give me an easy example of a pointwise convergent sequence which has a uniformly convergent subsequenice?
                      $endgroup$
                      – suchanda adhikari
                      Dec 31 '18 at 18:39










                    • $begingroup$
                      @suchandaadhikari That's easy. The sequence $0,0,0,dots$ will work.
                      $endgroup$
                      – zhw.
                      Dec 31 '18 at 18:40










                    • $begingroup$
                      nice one ..can you give me an example of a sequence of function which is not uniformly convergent but pointwise convergent and has a uniformly convergent subsequence please?
                      $endgroup$
                      – suchanda adhikari
                      Dec 31 '18 at 18:55






                    • 1




                      $begingroup$
                      @suchandaadhikari $x,0,x^2,0,x^3,0,dots $
                      $endgroup$
                      – zhw.
                      Dec 31 '18 at 19:03
















                    $begingroup$
                    can anyone give me an easy example of a pointwise convergent sequence which has a uniformly convergent subsequenice?
                    $endgroup$
                    – suchanda adhikari
                    Dec 31 '18 at 18:39




                    $begingroup$
                    can anyone give me an easy example of a pointwise convergent sequence which has a uniformly convergent subsequenice?
                    $endgroup$
                    – suchanda adhikari
                    Dec 31 '18 at 18:39












                    $begingroup$
                    @suchandaadhikari That's easy. The sequence $0,0,0,dots$ will work.
                    $endgroup$
                    – zhw.
                    Dec 31 '18 at 18:40




                    $begingroup$
                    @suchandaadhikari That's easy. The sequence $0,0,0,dots$ will work.
                    $endgroup$
                    – zhw.
                    Dec 31 '18 at 18:40












                    $begingroup$
                    nice one ..can you give me an example of a sequence of function which is not uniformly convergent but pointwise convergent and has a uniformly convergent subsequence please?
                    $endgroup$
                    – suchanda adhikari
                    Dec 31 '18 at 18:55




                    $begingroup$
                    nice one ..can you give me an example of a sequence of function which is not uniformly convergent but pointwise convergent and has a uniformly convergent subsequence please?
                    $endgroup$
                    – suchanda adhikari
                    Dec 31 '18 at 18:55




                    1




                    1




                    $begingroup$
                    @suchandaadhikari $x,0,x^2,0,x^3,0,dots $
                    $endgroup$
                    – zhw.
                    Dec 31 '18 at 19:03




                    $begingroup$
                    @suchandaadhikari $x,0,x^2,0,x^3,0,dots $
                    $endgroup$
                    – zhw.
                    Dec 31 '18 at 19:03











                    0












                    $begingroup$

                    Consider the sequnce $(f_n)_{ninmathbb N}$ of functions from $mathbb R$ into itself defined by$$f_n(x)=begin{cases}dfrac1{(1+x^2)^n}&text{ if }xneq0\0&text{ otherwise.}end{cases}$$and prove that no subsequence of this sequence converges uniformly. However, $(f_n)_{ninmathbb N}$ converges pointwise to the null function.






                    share|cite|improve this answer











                    $endgroup$













                    • $begingroup$
                      so you mean there is no possibility to get a uniformly convergent subsequence from any sequence of function which converges pointwise to zero?
                      $endgroup$
                      – suchanda adhikari
                      Dec 31 '18 at 18:23










                    • $begingroup$
                      Not at all. I only claimed that for the specific sequence that I defined.
                      $endgroup$
                      – José Carlos Santos
                      Dec 31 '18 at 18:28










                    • $begingroup$
                      Correct, however there are theorems about getting uniform convergence outside a set of small measure.
                      $endgroup$
                      – Charlie Frohman
                      Dec 31 '18 at 18:28










                    • $begingroup$
                      en.m.wikipedia.org/wiki/Egorov%27s_theorem
                      $endgroup$
                      – Charlie Frohman
                      Dec 31 '18 at 18:29
















                    0












                    $begingroup$

                    Consider the sequnce $(f_n)_{ninmathbb N}$ of functions from $mathbb R$ into itself defined by$$f_n(x)=begin{cases}dfrac1{(1+x^2)^n}&text{ if }xneq0\0&text{ otherwise.}end{cases}$$and prove that no subsequence of this sequence converges uniformly. However, $(f_n)_{ninmathbb N}$ converges pointwise to the null function.






                    share|cite|improve this answer











                    $endgroup$













                    • $begingroup$
                      so you mean there is no possibility to get a uniformly convergent subsequence from any sequence of function which converges pointwise to zero?
                      $endgroup$
                      – suchanda adhikari
                      Dec 31 '18 at 18:23










                    • $begingroup$
                      Not at all. I only claimed that for the specific sequence that I defined.
                      $endgroup$
                      – José Carlos Santos
                      Dec 31 '18 at 18:28










                    • $begingroup$
                      Correct, however there are theorems about getting uniform convergence outside a set of small measure.
                      $endgroup$
                      – Charlie Frohman
                      Dec 31 '18 at 18:28










                    • $begingroup$
                      en.m.wikipedia.org/wiki/Egorov%27s_theorem
                      $endgroup$
                      – Charlie Frohman
                      Dec 31 '18 at 18:29














                    0












                    0








                    0





                    $begingroup$

                    Consider the sequnce $(f_n)_{ninmathbb N}$ of functions from $mathbb R$ into itself defined by$$f_n(x)=begin{cases}dfrac1{(1+x^2)^n}&text{ if }xneq0\0&text{ otherwise.}end{cases}$$and prove that no subsequence of this sequence converges uniformly. However, $(f_n)_{ninmathbb N}$ converges pointwise to the null function.






                    share|cite|improve this answer











                    $endgroup$



                    Consider the sequnce $(f_n)_{ninmathbb N}$ of functions from $mathbb R$ into itself defined by$$f_n(x)=begin{cases}dfrac1{(1+x^2)^n}&text{ if }xneq0\0&text{ otherwise.}end{cases}$$and prove that no subsequence of this sequence converges uniformly. However, $(f_n)_{ninmathbb N}$ converges pointwise to the null function.







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited Dec 31 '18 at 18:27

























                    answered Dec 31 '18 at 18:04









                    José Carlos SantosJosé Carlos Santos

                    164k22131234




                    164k22131234












                    • $begingroup$
                      so you mean there is no possibility to get a uniformly convergent subsequence from any sequence of function which converges pointwise to zero?
                      $endgroup$
                      – suchanda adhikari
                      Dec 31 '18 at 18:23










                    • $begingroup$
                      Not at all. I only claimed that for the specific sequence that I defined.
                      $endgroup$
                      – José Carlos Santos
                      Dec 31 '18 at 18:28










                    • $begingroup$
                      Correct, however there are theorems about getting uniform convergence outside a set of small measure.
                      $endgroup$
                      – Charlie Frohman
                      Dec 31 '18 at 18:28










                    • $begingroup$
                      en.m.wikipedia.org/wiki/Egorov%27s_theorem
                      $endgroup$
                      – Charlie Frohman
                      Dec 31 '18 at 18:29


















                    • $begingroup$
                      so you mean there is no possibility to get a uniformly convergent subsequence from any sequence of function which converges pointwise to zero?
                      $endgroup$
                      – suchanda adhikari
                      Dec 31 '18 at 18:23










                    • $begingroup$
                      Not at all. I only claimed that for the specific sequence that I defined.
                      $endgroup$
                      – José Carlos Santos
                      Dec 31 '18 at 18:28










                    • $begingroup$
                      Correct, however there are theorems about getting uniform convergence outside a set of small measure.
                      $endgroup$
                      – Charlie Frohman
                      Dec 31 '18 at 18:28










                    • $begingroup$
                      en.m.wikipedia.org/wiki/Egorov%27s_theorem
                      $endgroup$
                      – Charlie Frohman
                      Dec 31 '18 at 18:29
















                    $begingroup$
                    so you mean there is no possibility to get a uniformly convergent subsequence from any sequence of function which converges pointwise to zero?
                    $endgroup$
                    – suchanda adhikari
                    Dec 31 '18 at 18:23




                    $begingroup$
                    so you mean there is no possibility to get a uniformly convergent subsequence from any sequence of function which converges pointwise to zero?
                    $endgroup$
                    – suchanda adhikari
                    Dec 31 '18 at 18:23












                    $begingroup$
                    Not at all. I only claimed that for the specific sequence that I defined.
                    $endgroup$
                    – José Carlos Santos
                    Dec 31 '18 at 18:28




                    $begingroup$
                    Not at all. I only claimed that for the specific sequence that I defined.
                    $endgroup$
                    – José Carlos Santos
                    Dec 31 '18 at 18:28












                    $begingroup$
                    Correct, however there are theorems about getting uniform convergence outside a set of small measure.
                    $endgroup$
                    – Charlie Frohman
                    Dec 31 '18 at 18:28




                    $begingroup$
                    Correct, however there are theorems about getting uniform convergence outside a set of small measure.
                    $endgroup$
                    – Charlie Frohman
                    Dec 31 '18 at 18:28












                    $begingroup$
                    en.m.wikipedia.org/wiki/Egorov%27s_theorem
                    $endgroup$
                    – Charlie Frohman
                    Dec 31 '18 at 18:29




                    $begingroup$
                    en.m.wikipedia.org/wiki/Egorov%27s_theorem
                    $endgroup$
                    – Charlie Frohman
                    Dec 31 '18 at 18:29


















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