Limits of an Integral: $\lim_{n\to \infty} \int x^2 \sin\left(f_n(x)\right)\mathrm{d}x=0$

Question

I have a problem with the following exercise. I think by the Lebesgue dominated convergence theorem it can be solved but i don't know that $\sin\left(f_n(x)\right) x^2$ dominated by ? It it true to this theorem? or not ? . I'm glad about every help. Assume that fn are integrable on R and fn→0 almost everywhere, as $n\to\infty$. Prove that for any $a>0$ : $$\lim_{n\to \infty} \int_{-a}^a x^2 \sin\left(f_n(x)\right)\mathrm{d}x=0$$

Answer 1

Use the inequality $$ \lvert x^2\sin\left(f_n(x)\right)\rvert\leqslant x^2 $$ and the function $x\mapsto x^2$ is integrable on the interval $(-a,a)$.