Let $f_n\in L^1(0,1)$ and $C>0$ be such that $f_n \geq 0, f_n \rightarrow 0$ a.e., and $$\int_0^1 \max\{f_1, ..., f_n\} dx \leq C \quad \text{ for every } n.$$ Prove that $f_n \rightarrow 0$ in $L^1(0,1)$.
I was working on some exams from the past. This question stood out because it seemed very simple. So I am doubting myself if I did it correctly.
Proof: Using Fatou's lemma, we have $$\int_0^1 \liminf_n \max\{f_1, ..., f_n\} dx \leq \liminf_n \int_0^1 \max\{f_1, ..., f_n\} dx \leq C,$$ therefore $\liminf_n \max\{f_1, ..., f_n\} $ is integrable. And we see that for each $f_n$, we have $$0\leq f_n \leq \liminf_n \max\{f_1, ..., f_n\} $$ by Dominated Convergence theorem we have $$\lim_n \int_0^1 f_n dx = 0.$$
Thanks a lot!
The proof looks correct. It can be made a little bit "nicer" if we notice and use the equality $\liminf_{n\to \infty}\max\{f_1,\dots,f_n\}=\sup_j f_j$ (hence we have much more than dominated convergence).