showing something is not poitnwise convergent if it diverges? say we had $f_n(x) = (-1)^n$ for $ x \in [0,1/n]$ and $f_n(x) = 0$ for $x \in (1/n,1]$. How would I explain that this is not pointwise convergent? Is it sufficient to say that since taking $n \to \infty$ we get that at $x = 0$ we can have the limit to be either $ 1 $ or $-1$? Similary, what if we had $(-1)^n n $ instead of $(-1)^n$

It is sufficient to show that a sequence of functions is not convergent at a single point to demonstrate that they do not converge pointwise. So the answer to your question is yes, that is enough.
To argue that $(-1)^n$ diverges, a simple way would be to demonstrate that it has two subsequential limits; $(-1)$ and $1$. A convergent sequence can have only one subsequential limit.
For the sequence $(-1)^n n$, a convergent sequence is a bounded sequence. Thus this is not convergent either.
