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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.

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Perhaps it would be helpful to first sketch the graphs of the first few $f_n$. –  David Mitra Dec 6 '12 at 12:26
    
@TUMO: It seems to be a CSIR-NET problem. –  Sugata Adhya Dec 6 '12 at 12:42

1 Answer 1

I think only 4. is correct. This you can prove by choosing any $x \in [0,1]$. If $x=0$ then $f_n(0)=1$ for all $n$. If $x>0$ then for all $n > \frac{1}{x}$ f_n(x)=0$. This is one of the standard examples on pointwise but non-uniform convergence, and is probably covered in most analysis courses...

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