Show $\displaystyle f(x) = \sum_{n=1}^{\infty}$ $\sin\left(\frac{x}{n^2}\right) \cos(nx)$, is continuous on $\mathbb{R}$.
Proof:
Let $r \gt 0$ be some fixed number. Then for $x \in [-r, r]$, $\sin\left(\frac{x}{n^2}\right) \cos(nx) \leq \frac{r}{n^2}$, since $\sin(a) \leq |a|$ and $\cos(b) \leq 1$.
So define $M_n = \frac{r}{n^2}$, then $\sum_n M_n \lt \infty$, so by Weierstrass M test, $f$ uniformly converges on $[-r,r]$ and hence, continuous.
Since $r$ was arbitrary, $f$ is continuous on $\mathbb{R}$.
Edit: I would like proof verification and some feedback on my proof. Possibly a better proof.