Suppose $f: [a,b] \to \mathbb{R}$ and $B$ satisfy $|f(x)| \le B$ for every $x\in [a,b]$.
Show that if $P = \{x_{0},...,x_{n}\}$ is a partition of $[a,b]$, then
$M(f^{2},[x_{i-1},x_{i}]) - m(f^{2},[x_{i-1},x_{i}]) \le 2B(M(f,[x_{i-1},x_{i}]) - m(f,[x_{i-1},x_{i}])) $
for every $1 \le i \le n$.
We are given a hint, namely that
$|f(x)^{2} - f(y)^{2}| = |f(x) - f(y)||f(x) + f(y)|$.
And this has something to do with Riemann integrals, or perhaps Darboux sums, as that is the section this homework was assigned in.