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:
- $\lim_{n\to\infty}f_n(x)$ defines a continuous function on $[0,1]$.
- $\{f_n\}$ converges uniformly on $[0,1]$.
- $\lim_{n\to\infty}f_n(x)=0$ for all $x\in [0,1]$.
- $\lim_{n\to\infty} f_n(x)$ exists for all $x\in[0,1]$.
I am completely stuck on it. Please help anyone.