Real Analysis quals I am working on this question: let $f_n$ Lebesgue measurable converge to $f$ in $L^1(\mathbb{R})$ norm and also assume $\|f_n\|_2\le M$ for all $n$ (that is the $L^2$ norm of all $f_n$ is bounded by some constant $M$). Show that:
1) $f\in L^2$
2) Does it follow that $f_n$ converges to $f$ in $L^2$
3) Show that $\|f-f_n\|_p\to 0$ for all $1<p<2$.
What I am doing is:
$|f|^2=|f-f_n+f_n|^2\le |f-f_n|^2 + 2|f_n||f-f_n| +|f_n|^2$
I know that if necessary I can pass to a subsequence of $f_n$ that converges pointwise a.e. to $f$; how can I reason for the first two terms? Any hints would be greatly appreciated.
For question 2 I think $f_n(x)=n\chi_{[0,1/n^2)}$ will give a counterexample.
I have no clue for 3. Any hints would also be greatly appreciated.

1) This is an application of Fatou's lemma. By passing to a subsequence $f_{n_k}$, we may assume $f_{n_k} \to f$ a.e. Now apply Fatou to the nonnegative sequence $f_{n_k}^2$. We have
$$\int f^2 dx \leq \liminf_{k\to \infty} \int f_{n_k}^2 \,dx \leq M^2.$$
2) Your counter example works here.
3) This is an application of Holder's inequality. Let $g=f-f_n$ and for $1<p<2$ write
$$|g|^p = |g|^{2p-2}|g|^{2-p}.$$
Apply Holder's inequality with exponents $p'=1/(p-1)$ and $q'=1/(2-p)$. (Check that $p',q'>1$ and $1/p' + 1/q'=1$). Then we have
$$\int |g|^p dx =\int |g|^{2p-2}|g|^{2-p} dx \leq \left(\int |g|^2 dx\right)^{p-1}\left(\int|g| dx\right)^{2-p}.$$
In other words
$$\|f-f_n\|_p^p \leq \|f-f_n\|_2^{2p-2}\|f-f_n\|_1^{2-p}\leq (2M)^{2p-2}\|f-f_n\|_1^{2-p}.$$
Since $f_n\to f$ in $L^1$, the estimate above gives convergence in $L^p$ for all $1<p<2$.
