Proving a sequence of functions converges, is differentiable, etc.












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I have this question asking to prove that a sequence of functions is differentiable, pointwise and uniformly convergent, and something regarding the equality of limits of the function's derivative.



For each natural number $n$, define $f_n : [−1, 1] → R$ by



$$f_n(x) = frac{x}{1+n^2x^2}$$



(a) Prove that for each natural number $n$, the function $f_n$ is differentiable, and find $f'_n$



(b) Prove that $(f_n)$ is uniformly convergent.



(c) Prove that $(f_n)$ is pointwise convergent.



(d) Prove that $$(lim_{xtoinfty} f_n(x))' neq lim_{xtoinfty} f_n'(x)$$



I notice that the function is continuous and that $f'_{n}$ exists over the interval, so that sets me up well to prove that $f_n$ is differentiable.



I'm iffy on proving the last three points. Where do I begin?






real-analysis uniform-convergence pointwise-convergence







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    0












    $begingroup$


    I have this question asking to prove that a sequence of functions is differentiable, pointwise and uniformly convergent, and something regarding the equality of limits of the function's derivative.



    For each natural number $n$, define $f_n : [−1, 1] → R$ by



    $$f_n(x) = frac{x}{1+n^2x^2}$$



    (a) Prove that for each natural number $n$, the function $f_n$ is differentiable, and find $f'_n$



    (b) Prove that $(f_n)$ is uniformly convergent.



    (c) Prove that $(f_n)$ is pointwise convergent.



    (d) Prove that $$(lim_{xtoinfty} f_n(x))' neq lim_{xtoinfty} f_n'(x)$$



    I notice that the function is continuous and that $f'_{n}$ exists over the interval, so that sets me up well to prove that $f_n$ is differentiable.



    I'm iffy on proving the last three points. Where do I begin?






    real-analysis uniform-convergence pointwise-convergence







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      I have this question asking to prove that a sequence of functions is differentiable, pointwise and uniformly convergent, and something regarding the equality of limits of the function's derivative.



      For each natural number $n$, define $f_n : [−1, 1] → R$ by



      $$f_n(x) = frac{x}{1+n^2x^2}$$



      (a) Prove that for each natural number $n$, the function $f_n$ is differentiable, and find $f'_n$



      (b) Prove that $(f_n)$ is uniformly convergent.



      (c) Prove that $(f_n)$ is pointwise convergent.



      (d) Prove that $$(lim_{xtoinfty} f_n(x))' neq lim_{xtoinfty} f_n'(x)$$



      I notice that the function is continuous and that $f'_{n}$ exists over the interval, so that sets me up well to prove that $f_n$ is differentiable.



      I'm iffy on proving the last three points. Where do I begin?






      real-analysis uniform-convergence pointwise-convergence







      share|cite|improve this question









      $endgroup$




      I have this question asking to prove that a sequence of functions is differentiable, pointwise and uniformly convergent, and something regarding the equality of limits of the function's derivative.



      For each natural number $n$, define $f_n : [−1, 1] → R$ by



      $$f_n(x) = frac{x}{1+n^2x^2}$$



      (a) Prove that for each natural number $n$, the function $f_n$ is differentiable, and find $f'_n$



      (b) Prove that $(f_n)$ is uniformly convergent.



      (c) Prove that $(f_n)$ is pointwise convergent.



      (d) Prove that $$(lim_{xtoinfty} f_n(x))' neq lim_{xtoinfty} f_n'(x)$$



      I notice that the function is continuous and that $f'_{n}$ exists over the interval, so that sets me up well to prove that $f_n$ is differentiable.



      I'm iffy on proving the last three points. Where do I begin?






      real-analysis uniform-convergence pointwise-convergence




      real-analysis uniform-convergence pointwise-convergence






      share|cite|improve this question













      share|cite|improve this question











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      asked Jan 15 at 4:37









      beflyguybeflyguy

      354




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          $begingroup$

          $(b)$: prove that $|f_n(x)| le frac{1}{2n}$ for all $n in mathbb N$ and all $x in [0,1].$



          $(c)$ follows from $(b)$.



          $(d)$ is now your turn !






          share|cite|improve this answer









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            1 Answer
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            active

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            1












            $begingroup$

            $(b)$: prove that $|f_n(x)| le frac{1}{2n}$ for all $n in mathbb N$ and all $x in [0,1].$



            $(c)$ follows from $(b)$.



            $(d)$ is now your turn !






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              $(b)$: prove that $|f_n(x)| le frac{1}{2n}$ for all $n in mathbb N$ and all $x in [0,1].$



              $(c)$ follows from $(b)$.



              $(d)$ is now your turn !






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                $(b)$: prove that $|f_n(x)| le frac{1}{2n}$ for all $n in mathbb N$ and all $x in [0,1].$



                $(c)$ follows from $(b)$.



                $(d)$ is now your turn !






                share|cite|improve this answer









                $endgroup$



                $(b)$: prove that $|f_n(x)| le frac{1}{2n}$ for all $n in mathbb N$ and all $x in [0,1].$



                $(c)$ follows from $(b)$.



                $(d)$ is now your turn !







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 15 at 5:46









                FredFred

                46.8k1848




                46.8k1848






























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