Exercise 4.2.7

Let $g:A\to R$ and assume that $f$ is a bounded function on $A$ in the sense that there exists $M>0$ satisfying $|f(x)|\le M$ for all $x\in A$. Show that if $\underset{x\to c}{\lim <!--\; FUNCTION\; APPLICATION\; -->}g(x)=0$, then $\underset{x\to c}{\lim <!--\; FUNCTION\; APPLICATION\; -->}g(x)f(x)=0$ as well.