4. (6 points) Prove the following fact that was stated without proof in lecture:
Let $f$ be a bounded function on $[a, b]$. Then the following are equivalent:
(i) $f$ is Riemann integrable.
(ii) For every $\epsilon > 0$, there exists a partition $P$ of $[a, b]$ such that
$\overline{S}(f, P) - \underline{S}(f, P) < \epsilon$.
(iii) For every $\epsilon > 0$, there exist partitions $P$ and $Q$ of $[a, b]$ such that
$\overline{S}(f, P) - \underline{S}(f, Q) < \epsilon$.
