Originally Posted by

**spoon737** This problem is giving me a headache.

Suppose $\displaystyle \{f_n\}$ is a sequence of functions defined on [a,b] which converges uniformly to f. Prove that $\displaystyle \{f_n\}$ is equicontinuous. That is, show that whenever $\displaystyle \epsilon >0$, there is a $\displaystyle \delta >0$ such that if n is a positive integer and $\displaystyle x,y \in [a,b]$ with $\displaystyle |x-y|< \delta$, then $\displaystyle |f_n(x)-f_n(y)|< \epsilon$.

Any help on how to prove this would be greatly appreciated. Bear in mind that this is the first time I've heard of equicontinuity, so if there is some nice theorem that produces this as an immediate result, I probably don't know it and thus can't use it. I need to work with the epsilon-delta definition.