If $f_n \rightarrow 0$ and $\int \sup f_1, ... , f_n \leq M$ then $\int f_n \rightarrow 0$

Question

Let $f_n$ be a sequence of nonnegative measurable functions which converge to $0$. If there exists an $M$ such that $$\int \sup f_1, ... , f_n \leq M$$ for all $n$, then $\lim \int f_n = 0$. Could someone give me a hint on this? I don't know how to start.

Answer 1

This follows via an application of the dominated convergence theorem.

Answer 2

Let $h_n = \sup\{f_1,\ldots,f_n\}$ and suppose $h_n$ increases to $f$. Then $|f_n| = f_n \le h_n \le f$ and by the monotone convergence theorem,

$$\int f = \lim_n \int h_n \le M.$$

Hence, by the dominated convergence theorem, $\int f_n \to 0$.