# Prove that $\lim_{n\to\infty}\int f_n=\int f$.

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.

Your proof is correct! Depending on the level of detail you want to include you might want to consider adding a sentence on why we can deduce that $$\lim_{n \to \infty} \int f_n$$ exists from the fact that $$\limsup \int f_n \leq \liminf \int f_n$$.