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Suppose each $f_n$ is continuous on $S$ and that $f=\lim_{n\rightarrow\infty} f_n$ is also continuous on $S$. Does this imply that $f_n$ converges uniformly on $S$?

I know that the uniform limit of continuous functions is continuous. However, is the converse true?

CoffeeIsLife
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1 Answers1

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No and for the counter example let $f_n(x)=x^n, \; x\in[0,1)$ we have $f_n$ doesn't converge uniformly to the zero function since $||f_n||_\infty=1$.

  • I think I will add more to this, just in case it is helpful down the line. I think the reason why Sami Ben Romdhane says that $\vert\vert f_n\vert\vert_{\infty}=1$ proves that $f_n$ doesn't converge uniformly to $f$ is because of a theorem/remark. According to the remark/theorem, a sequence $(f_n)$ of functions on a set $S$ converges uniformly to a function $f$ on $S$ if and only if $$\lim_{n\rightarrow\infty}\sup{\vert f(x)-f_n(x)\vert :x\in S}=0$$ – CoffeeIsLife Dec 13 '13 at 20:15
  • Here, $$\lim_{n\rightarrow\infty}\sup{\vert f(x)-f_n(x)\vert :x\in S}=0$$

    Substituting $f(x)=0$ and we are left with $\lim_{n\rightarrow\infty}{\vert f_n\vert :x\in S\vert}=1\neq 0$

     – CoffeeIsLife Dec 13 '13 at 20:16