Language around Pointwise convergence v. Uniform convergence I've heard uniform convergence described loosely as convergence "occurring at the same rate at every point." Is that really accurate? If the absolute difference between a member of a sequence of functions and the function to which that sequence converges pointwise can be represented by a function of the point with a finite maximum, then isn't it true that the sequence of functions uniformly convergent AND the rate of convergence varies with the point?

Given an $\epsilon>0$ and $x$, we can define $N_x(\epsilon)$ as the smallest $N$ such that $|f_n(x)-f(x)|<\epsilon$ for all $n\geq N.$
In some sense, $N_x(\epsilon)$ is a measure of the rate convergence at $x.$ 
Uniform convergence means that for any $\epsilon>0,$ $N(\epsilon)=\sup_{x} N_x(\epsilon)<+\infty.$
So this doesn't strictly mean they converge at the same speed, but that there is some speed of convergence which is slower than (or equal to) all of them.
