$\epsilon$ rules for uniform and pointwise convergence.

Question

Could someone please provide with the $\epsilon$ definitions of uniform and pointwise convergence. I'm trying to really get my head around the differences between them (I do know the differences, but it would help to see them written formally in this way).

Answer 1

Let $(f_n)$ be a sequence of functions $f_n:X\rightarrow Y$, where $X$ is a set and $Y$ a metric space. The sequence $(f_n)$ converges uniformly to $f$ if for every $\epsilon>0$ there exists an $N$, such that for all $n\ge N$ and for every $x\in X$ : $d(f(x),f_n(x))<\epsilon$. Pointwise convergence is the same, but now you can choose $N$ depending on $x$.