Uniform Convergence: The two definitions I know of the two equivalent definitions for uniform convergence. Namely:
$f_n(x)$ converges uniformly to $f(x)$ if either:
a)$$ \forall \epsilon \exists N \in \mathbb N s.t. \forall x \in D \ \forall n \geq N:
|f_n(x) - f| < \epsilon $$
or
b) $$\lim_{n \rightarrow \infty} \sup_{x\in D} |f_n(x) - f(x)| = 0 $$
While I see why in the case of a) the same "speed of convergence" is guaranteed since one epsilon is chosen for all $x$, I unfortunately cannot make sense of b) on an intuitive level. So I see that b) states that the biggest difference between the sequences of the functions and the limit functions must become zero for n going to infinity. But how does that also guarantee this convergence process is of "uniform speed" for all x?

The statement that the limit exists in (b) is exactly the statement that $\forall \; \epsilon > 0, \exists \; n \; \text{ s.t. } \forall m > n, \sup |f_m - f| < \epsilon$. The 'uniform' part, i.e. the for all $x$ part, comes as the $\sup$ naturally pulls out the biggest offender.
