real analysis - Uniformly convergent implies equicontinuous

I'm trying to prove that if I have a sequence of continuously differentiable functions $f_n$ that converge uniformly on $[a,b]$, then $\{f_n\}$ is equicontinuous.

My idea is to use uniform convergence to deal with the "tail" and then use continuity to deal with the finitely many $f_n$'s left. But I'm having trouble writing it down...