Let $f_n: \mathbb{R} \rightarrow \mathbb{R}$ for all $n \in \mathbb{N}$ defined by $f_n (x) := n \sin (\frac{x}{n})$. Prove the sequence is equicontinuous.
I know it's just a $\delta - \epsilon$ proof, but I think my trig is just too rusty. I can't seem to manipulate it in any useful way.
$$|f_n(x)-f_n(y)|=n| \sin (x/n)-\sin (y/n)|$$ $$= n (1/n) |\cos (\psi/n)||x-y|\leq |x-y|$$ using Mean Value Theorem. The above is true for all $x,y$ and for all $n$. In particular this implies uniform equicontinuity (the analogous of uniform continuity) which is stronger than equicontinuity.