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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.

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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 $\limsup \int f_n \leq \liminf \int f_n$.

For future reference, the result you have proved is called Lebesgue's Monotone Convergence Theorem. It is an interesting fact that one can prove the other implication as well, that is, one can derive Fatou's Lemma assuming Lebesgue's Monotone Convergence Theorem. So, if you can write down a proof of LMCT without invoking Fatou's Lemma, then you will have shown that these two results are equivalent to each other. You can take a look at Rudin's Real and Complex Analysis for such a proof of LMCT.

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