let $E$ subset of $\mathbb R$ be compact and the function $f$ is continuous on $E$ and the function $g$ is integrable on $E$. prove that the function $fg$ is integrable.
A continuous function $f$ on a compact set $E$ of $\mathbf R$ is bounded; that is, the sup-norm of $f$ is finite (i.e., $\|f\|_\infty < \infty$).
Now note that
$$\int |fg| \leqslant \|f\|_\infty \int |g| < \infty.$$
Also, follow t.b.'s advice please.