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Define $$ f_n(x)= \begin{cases} 1-nx, &x\in [0,1/n],\\ 0, &x\in [1/n,1] \end{cases} $$

Then which of the following is correct:

  1. $\lim_{n\to\infty}f_n(x)$ defines a continuous function on $[0,1]$.
  2. $\{f_n\}$ converges uniformly on $[0,1]$.
  3. $\lim_{n\to\infty}f_n(x)=0$ for all $x\in [0,1]$.
  4. $\lim_{n\to\infty} f_n(x)$ exists for all $x\in[0,1]$.

I am completely stuck on it. Please help anyone.

  • 0
    Perhaps it would be helpful to first sketch the graphs of the first few $f_n$.2012-12-06
  • 0
    @TUMO: It seems to be a CSIR-NET problem.2012-12-06

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