Uniform & Pointwise convergence of $\frac{\sin(nx)}{n}$ and $\frac{x\cos(nx)}{n}$

Question

Here are some exercises I've found:

Prove or disprove it is uniform and/or pointwise convergent for the following:

1. $f_n(x)=\frac{sin(nx)}{n}$ defined on $(0,+\infty)$
2. $f_n(x)=\frac{x\cos(nx)}{n}$ defined on $(0,+\infty)$

Proof 1: My thoughts are that $f_n$ is pointwise with $f(x)=\lim_{n \to \infty}f_n(x)=0$. We can then prove uniform convergence using the supremum test

$$ \sup_{t \in (0,+\infty)}\left|\frac{\sin(nt)}{n}\right|=\frac{1}{n} \to 0 \text{ as } n \to \infty $$

Proof 2: It appears we have pointwise convergence with $\lim_{n \to \infty}f_n(x)=0$. However, uniform convergence doesn't hold by the supremum test

$$ \sup_{t \in (0,+\infty)}\left|\frac{t\cos(nt)}{n}\right|\to \infty \text{ as } n \to \infty $$

What are your thoughts on my method?