Let $f:[0,1] \rightarrow \mathbb{R}$ satisfy $|f(x)-f(y)| \leqslant|x-y|$ for all $x, y \in[0,1]$.

Show that $f$ is continuous and that for all $\varepsilon>0$, there exists a piecewise constant function $g$ such that

$$\sup _{x \in[0,1]}|f(x)-g(x)| \leqslant \varepsilon .$$

For all integers $n \geqslant 1$, let $u_{n}=\int_{0}^{1} f(t) \cos (n t) d t$. Show that the sequence $\left(u_{n}\right)$ converges to 0 .