I am reviewing for a qual and below is a question I do not know how to solve.
Let $(f_n)$ be a sequence of Lebsegue measurable functions on $\mathbb{R}$ that converge to $f$ in $L_1$. Suppose in addition that $f_n\in L^2$ and there exists a constant $M$ such that $\|f_n\|_2< M$. Prove that $f\in L^2$.
The following is my attempt.
We have $$\left(\int|f|^2\right)^{1/2}=\left(\int|f-f_n+f_n|^2\right)^{1/2}\leq \left(\int(|f-f_n|+|f_n|)^2\right)^{1/2}\leq 2\left(\int |f-f_n|^2+\int|f_n|^2\right)^{1/2},$$ where the last inequality follows from the inequality $(|f|+|g|)^p\leq 2^p(|f|^p+|g|^p)$. Now we know that for all $n$, $$\int |f_n|^2<M^2.$$ But I do not know how to use the assumption that $f_n\rightarrow f$ in $L^1$ to show that $$\int|f-f_n|^2$$ is bounded. Maybe there is another way to prove that $f\in L^2$, any help will be appreciated.
We can use the following facts:
If a sequence converges almost everywhere and is bounded in $\mathbb L^2$, then the limit belongs to $\mathbb L^2$, as a consequence of Fatou's lemma.
If a sequence converges in $\mathbb L^1$, then we can extract an almost everywhere convergent subsquence.