The sequence of functions $f_n = \sin{nx}$ is not uniformly equicontinuous on any compact interval.
I already proved that this sequence is not uniformly equicontinuous on $[0,1]$:
Let $f_n(x) = \sin(nx)$ be defined on $[0,1]$. Put $\epsilon = \sin(1)$. Let $\delta >0$. Choose $N \in \mathbb{N}$ such that $\frac{1}{N} < \delta$ and define $x_N = 1/N$ and $y_N = 0$. Then $|x_N-y_N| = 1/N < \delta$ while $$ |f_N(x_N) - f_N(y_N)| = |\sin(1) - \sin(0)| = |\sin(1)| \geq \epsilon. $$ This proves that $\{f_n\}$ is not uniformly equicontinuous on $[0,1]$.
My question is this: is there a way to use the fact that $[0,1]$ is homeomorphic to $[a,b]$ to prove the claim?