Suppose that $(\phi_n)$ is a sequence of step functions defined on $[a,b]$ and $f:[a,b]\rightarrow \mathbb{R}$ is such that \begin{equation*} \mathrm{sup}(|f(t)-\phi_n(t)|)\rightarrow0 \end{equation*} Show that f is Riemann integrable.
I have no idea where to start, any hints would be appreciated