My problem is that I find it kind of hard to contrast between uniform and pointwise convergence. For example with this proof I'm not quite sure whether I have proven uniform or poitwise convergence:
$ f_n(x) := \left\{\begin{array}{ll} 2n^2x, & x\in [0,\frac{1}{2n}) \\ -2n^2x+2n, & x\in [\frac{1}{2n},\frac{1}{n})\\ 0,& x \in [\frac{1}{n},1]\end{array}\right. . $
Claim: $(f_n)$ converges pointwise/ uniformly (im not sure) to $f$, where $f(x):= 0$
Proof: Let $\epsilon > 0$
Case one (x = 0):
$\mid f_n(0) -f(0)\mid = \mid 0-0\mid = 0 < \epsilon $ (by definition)
Case two ($x \in (0,1]$):
Set $N \in \mathbb{N}: \frac{1}{N} < x$, now when $n \geq N \Rightarrow \frac{1}{n} < x$ then by definition of $f_n$
$\Rightarrow f_n(x) = 0 \Rightarrow \mid f_n(x)-f(x) \mid = \mid 0-0\mid = 0 < \epsilon $ (by definiton)
Thus the claim is proven q.e.d
I'm asking which type of convergence is this now and whichever it is how would you show the other type?