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By Arzela-Ascoli, if I show that A is bounded and equicontinuous. Then, it is totally bounded. I have no problem showing that it is bounded. But I do not know how to determine it is equicontinuous or not. Can you help me? Thanks for any help.

Mahmut Esat Akın
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They are not equicontinuous. Otherwise, you would have $\delta>0$ such that for all $n\in \mathbb{N}$ and all $\vert x \vert < \delta$ holds $\vert \sin(n x) \vert =\vert \sin(nx) - \sin(0) \vert <\frac{1}{2}$, which is clearly not the case.

Severin Schraven
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