Let $(f_n)$ be a sequence of non-negative Lebesgue measurable functions such that $f_n$ converges to $f$ and $f_n\leq f\ \forall n\in \mathbb{Z^+}$. It is needed to prove that $\lim_{n\to\infty}\int f_n=\int f$. The following is my solution.
By Fatou's lemma $\int f=\int (\liminf f_n)\leq \liminf\int f_n$. Since $f_n\leq f\ \forall n\in \mathbb{Z^+}$, $\int f_n\leq\int f$. Therefore $\limsup\int f_n\leq \int f$. Therefore $\limsup\int f_n\leq \int f\leq \liminf\int f_n$. Hence $\lim_{n\to\infty}\int f_n=\int f$.
Could someone tell me if my proof is correct? Thanks.