sequence of function $f_n(x)= \sin(n\pi x)$

Question

$f_n(x):[0,1]\rightarrow \mathbb{R}$ defined by $$f_n(x)= \sin(n\pi x)$$ if $x\in [0,1/n]$, and $$f_n(x)=0$$ if $x\in (1/n,1]$ Then

1. It Does not converge pointwise.

2. It converge pointwise but the limit is not continous.

3. It converge pointwise but not uniformly.

4. It converge uniformly.

well, limit function is $0$ clearly, clearly $\sup|f_n(x)-0|\rightarrow 0$ as $n\rightarrow \infty$ and $x\in [0,1]$ so, it converges uniformly to $0$ am I right? so in my guess, 4 is only correct.

Answer 1

Here is a hint for your problem:

Note that given an $n$ you will always be able to find a $c \in [0, \frac{1}{n}]$ such that $f_n(c) = 1$.