Uniform convergence of $frac{y/(2N)}{sin(y/(2N))}$ towards 1












2














I can't come up with a proof, why $f_N(y) := frac{frac{y}{2N}}{sinleft(frac{y}{2N}right)}$ converges uniformly against $1$ for $yin(0,pi), Ntoinfty$.



I would be thankful for any advice.










share|cite|improve this question




















  • 4




    One idea is to use $x-x^3/6 leq sin(x) leq x$ which holds for $x geq 0$.
    – Ian
    Jan 4 at 15:15










  • Ian.Want to post an answer with your idea?
    – Peter Szilas
    Jan 4 at 15:32










  • What do you mean by $y/2N$? $frac{y}{2N}$ or $frac{y}{2}N$?
    – mathcounterexamples.net
    Jan 4 at 15:58










  • @mathcounterexamples.net For the result to be as they say it must be the former...
    – Ian
    Jan 4 at 16:02






  • 1




    @stressedout The poles are nowhere to be seen here, even when $N=1$, because of the $2$. $1-x^2/6 leq sin(x)/x leq 1$ simply becomes $1 leq x/sin(x) leq frac{1}{1-x^2/6}$ which is valid for $0<x<sqrt{6}$, and $pi/2<sqrt{6}$.
    – Ian
    Jan 4 at 16:10


















2














I can't come up with a proof, why $f_N(y) := frac{frac{y}{2N}}{sinleft(frac{y}{2N}right)}$ converges uniformly against $1$ for $yin(0,pi), Ntoinfty$.



I would be thankful for any advice.










share|cite|improve this question




















  • 4




    One idea is to use $x-x^3/6 leq sin(x) leq x$ which holds for $x geq 0$.
    – Ian
    Jan 4 at 15:15










  • Ian.Want to post an answer with your idea?
    – Peter Szilas
    Jan 4 at 15:32










  • What do you mean by $y/2N$? $frac{y}{2N}$ or $frac{y}{2}N$?
    – mathcounterexamples.net
    Jan 4 at 15:58










  • @mathcounterexamples.net For the result to be as they say it must be the former...
    – Ian
    Jan 4 at 16:02






  • 1




    @stressedout The poles are nowhere to be seen here, even when $N=1$, because of the $2$. $1-x^2/6 leq sin(x)/x leq 1$ simply becomes $1 leq x/sin(x) leq frac{1}{1-x^2/6}$ which is valid for $0<x<sqrt{6}$, and $pi/2<sqrt{6}$.
    – Ian
    Jan 4 at 16:10
















2












2








2







I can't come up with a proof, why $f_N(y) := frac{frac{y}{2N}}{sinleft(frac{y}{2N}right)}$ converges uniformly against $1$ for $yin(0,pi), Ntoinfty$.



I would be thankful for any advice.










share|cite|improve this question















I can't come up with a proof, why $f_N(y) := frac{frac{y}{2N}}{sinleft(frac{y}{2N}right)}$ converges uniformly against $1$ for $yin(0,pi), Ntoinfty$.



I would be thankful for any advice.







real-analysis 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 4 at 16:06









robjohn

265k27303625




265k27303625










asked Jan 4 at 14:59









SlyderSlyder

163




163








  • 4




    One idea is to use $x-x^3/6 leq sin(x) leq x$ which holds for $x geq 0$.
    – Ian
    Jan 4 at 15:15










  • Ian.Want to post an answer with your idea?
    – Peter Szilas
    Jan 4 at 15:32










  • What do you mean by $y/2N$? $frac{y}{2N}$ or $frac{y}{2}N$?
    – mathcounterexamples.net
    Jan 4 at 15:58










  • @mathcounterexamples.net For the result to be as they say it must be the former...
    – Ian
    Jan 4 at 16:02






  • 1




    @stressedout The poles are nowhere to be seen here, even when $N=1$, because of the $2$. $1-x^2/6 leq sin(x)/x leq 1$ simply becomes $1 leq x/sin(x) leq frac{1}{1-x^2/6}$ which is valid for $0<x<sqrt{6}$, and $pi/2<sqrt{6}$.
    – Ian
    Jan 4 at 16:10
















  • 4




    One idea is to use $x-x^3/6 leq sin(x) leq x$ which holds for $x geq 0$.
    – Ian
    Jan 4 at 15:15










  • Ian.Want to post an answer with your idea?
    – Peter Szilas
    Jan 4 at 15:32










  • What do you mean by $y/2N$? $frac{y}{2N}$ or $frac{y}{2}N$?
    – mathcounterexamples.net
    Jan 4 at 15:58










  • @mathcounterexamples.net For the result to be as they say it must be the former...
    – Ian
    Jan 4 at 16:02






  • 1




    @stressedout The poles are nowhere to be seen here, even when $N=1$, because of the $2$. $1-x^2/6 leq sin(x)/x leq 1$ simply becomes $1 leq x/sin(x) leq frac{1}{1-x^2/6}$ which is valid for $0<x<sqrt{6}$, and $pi/2<sqrt{6}$.
    – Ian
    Jan 4 at 16:10










4




4




One idea is to use $x-x^3/6 leq sin(x) leq x$ which holds for $x geq 0$.
– Ian
Jan 4 at 15:15




One idea is to use $x-x^3/6 leq sin(x) leq x$ which holds for $x geq 0$.
– Ian
Jan 4 at 15:15












Ian.Want to post an answer with your idea?
– Peter Szilas
Jan 4 at 15:32




Ian.Want to post an answer with your idea?
– Peter Szilas
Jan 4 at 15:32












What do you mean by $y/2N$? $frac{y}{2N}$ or $frac{y}{2}N$?
– mathcounterexamples.net
Jan 4 at 15:58




What do you mean by $y/2N$? $frac{y}{2N}$ or $frac{y}{2}N$?
– mathcounterexamples.net
Jan 4 at 15:58












@mathcounterexamples.net For the result to be as they say it must be the former...
– Ian
Jan 4 at 16:02




@mathcounterexamples.net For the result to be as they say it must be the former...
– Ian
Jan 4 at 16:02




1




1




@stressedout The poles are nowhere to be seen here, even when $N=1$, because of the $2$. $1-x^2/6 leq sin(x)/x leq 1$ simply becomes $1 leq x/sin(x) leq frac{1}{1-x^2/6}$ which is valid for $0<x<sqrt{6}$, and $pi/2<sqrt{6}$.
– Ian
Jan 4 at 16:10






@stressedout The poles are nowhere to be seen here, even when $N=1$, because of the $2$. $1-x^2/6 leq sin(x)/x leq 1$ simply becomes $1 leq x/sin(x) leq frac{1}{1-x^2/6}$ which is valid for $0<x<sqrt{6}$, and $pi/2<sqrt{6}$.
– Ian
Jan 4 at 16:10












1 Answer
1






active

oldest

votes


















0














A solution using Dini's theorem



The $f_n$ can be extended by continuity at $0$ by raising $f_n(0)=1$. Hence, we can consider the continuous extended maps on the compact interval $[0,pi]$.



For $n ge 2$ the sequence $(f_n)$ is uniformly bounded below by a constant strictly positive. Hence proving the uniform convergence of $(f_n)$ is equivalent to the proof of the uniform convergence of $(1/f_n)$. And this is provided by Dini's theorem as for all $x in [0, pi]$ the sequence $(1/f_n(x))$ is increasing.






share|cite|improve this answer





















    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%2f3061733%2funiform-convergence-of-fracy-2n-siny-2n-towards-1%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0














    A solution using Dini's theorem



    The $f_n$ can be extended by continuity at $0$ by raising $f_n(0)=1$. Hence, we can consider the continuous extended maps on the compact interval $[0,pi]$.



    For $n ge 2$ the sequence $(f_n)$ is uniformly bounded below by a constant strictly positive. Hence proving the uniform convergence of $(f_n)$ is equivalent to the proof of the uniform convergence of $(1/f_n)$. And this is provided by Dini's theorem as for all $x in [0, pi]$ the sequence $(1/f_n(x))$ is increasing.






    share|cite|improve this answer


























      0














      A solution using Dini's theorem



      The $f_n$ can be extended by continuity at $0$ by raising $f_n(0)=1$. Hence, we can consider the continuous extended maps on the compact interval $[0,pi]$.



      For $n ge 2$ the sequence $(f_n)$ is uniformly bounded below by a constant strictly positive. Hence proving the uniform convergence of $(f_n)$ is equivalent to the proof of the uniform convergence of $(1/f_n)$. And this is provided by Dini's theorem as for all $x in [0, pi]$ the sequence $(1/f_n(x))$ is increasing.






      share|cite|improve this answer
























        0












        0








        0






        A solution using Dini's theorem



        The $f_n$ can be extended by continuity at $0$ by raising $f_n(0)=1$. Hence, we can consider the continuous extended maps on the compact interval $[0,pi]$.



        For $n ge 2$ the sequence $(f_n)$ is uniformly bounded below by a constant strictly positive. Hence proving the uniform convergence of $(f_n)$ is equivalent to the proof of the uniform convergence of $(1/f_n)$. And this is provided by Dini's theorem as for all $x in [0, pi]$ the sequence $(1/f_n(x))$ is increasing.






        share|cite|improve this answer












        A solution using Dini's theorem



        The $f_n$ can be extended by continuity at $0$ by raising $f_n(0)=1$. Hence, we can consider the continuous extended maps on the compact interval $[0,pi]$.



        For $n ge 2$ the sequence $(f_n)$ is uniformly bounded below by a constant strictly positive. Hence proving the uniform convergence of $(f_n)$ is equivalent to the proof of the uniform convergence of $(1/f_n)$. And this is provided by Dini's theorem as for all $x in [0, pi]$ the sequence $(1/f_n(x))$ is increasing.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 4 at 16:17









        mathcounterexamples.netmathcounterexamples.net

        25.3k21953




        25.3k21953






























            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.





            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.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3061733%2funiform-convergence-of-fracy-2n-siny-2n-towards-1%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

            1300-talet

            1300-talet

            Display a custom attribute below product name in the front-end Magento 1.9.3.8