Let $D$ be compact, connected. Let $f$ and $g$ be continuous and bounded on $D$,with $g(p) \ge 0$ for all $p \in D$. Then, there is a point $p_0 \in D$ such that $\iint_D fg = f(p_0)\iint_D g$.
The hints from the back of the book say that I'm supposed to start off by showing that $\iint_D fg$ lies between $M \iint_D g$ and $m\iint_D g$, where $m$ and $M$ denote the inf and sup of $f$, respectively. Since $mg \le fg \le Mg$, am I just supposed to apply the double integral to the inequality to get that $m \iint_D g \le \iint_D fg \le M\iint_D g$?