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












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












    $begingroup$


    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






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 31 '18 at 18:00









      suchanda adhikarisuchanda adhikari

      586




      586






















          3 Answers
          3






          active

          oldest

          votes


















          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









          $endgroup$





















            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











            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
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3057911%2fexample-of-a-uniformly-convergent-subsequence-of-a-sequence-of-function-which-is%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









            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


















                    draft saved

                    draft discarded




















































                    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.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3057911%2fexample-of-a-uniformly-convergent-subsequence-of-a-sequence-of-function-which-is%23new-answer', 'question_page');
                    }
                    );

                    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







                    -

                    Popular posts from this blog

                    Quarter-circle Tiles

                    Image
                    up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2
                    Read more

                    build a pushdown automaton that recognizes the reverse language of a given pushdown automaton?

                    Image
                    0 I think it's similiar to NFAs. I replace the finite states of the given automaton for start-states for my new automaton. I do it with $epsilon$ -transition from the start state to the actual finite states of the given automaton. The start state of the given automaton will be my accepting(finite) state of my new one. All transitions will be reversed of course to recognize the reversed word. But how do I have to change the stack of the pushdown automaton? That's tricky. formal-languages automata context-free-grammar share | cite | improve this question edited Nov 29 '18 at 14:20 joee
                    Read more

                    Mont Emei

                    Image
                    Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog
                    Read more