How can I prove that the product of a sequence of a function and a function is integrable?

For clarity I will give a complete proof of the statement. First you need to know, or prove, that

$$ \left| \int_{a}^b h(x)dx \right|\leqslant \int_{a}^b |h(x)|dx $$

when the last integral is finite. Also, as $f$ is continuous and it domain is compact then the image of $f$ is also compact, this imply that the image of $f$ is bounded, say by a constant $M>0$, therefore

$$ 0\leqslant \left| \int_{a}^b f(x)g_n(x) \right|\leqslant \int_{a}^b |f(x)g_n(x)| dx\leqslant M \int_{a}^b |g_n(x)|<\infty $$

for all $n$. As limits of real-valued sequences respect order relations, taking limits above you find that

$$ 0\leqslant \lim_{n\to\infty}\left|\int_{a}^{b}f(x)g_n(x)dx\right|\leqslant M\cdot 0=0\\ \therefore\quad \lim_{n\to\infty}\int_{a}^{b}f(x)g_n(x)dx=0 $$

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