I'm thinking about a sequence of functions on closed interval $[0,1]$, where $f_n(x)=(-1)^nn$ for$x \in (0,1/n]$, and $0$ elsewhere.
It's clear that $f_n$ is not uniformly convergent since the supremum doesn't go to zero. But does it converge pointwise? Intuitively, I think it does converge to $f(x)=0$, because as $n$ goes to infinity, the interval should vanish. But I have no idea how to prove it formally.
Could anyone tell me whether I'm correct, and how to prove this pointwise convergence?