1. Prove that if $f_n$ converges to $0$ in $L^2$ and $\|f_n\|_{L^2} \leq 1$ for each $n$, then $\frac{f_n(x)}{n}$ converges pointwise to $0$ a.e.
Proof: Pick $\epsilon > 0$. Let $A_n = \left\{ x : \left| \frac{f_n(x)}{n} \right| \geq \epsilon \right\}$. Then we have
$$
\begin{eqnarray}
\frac{1}{n^2} &\geq& \int \left| \frac{f_n}{n} \right|^2 \\
&\geq& \int_{A_n} \left |\frac{f_n}{n} \right|^2 \\
&\geq& \int_{A_n} \epsilon^2 \\
&=& \epsilon^2 m(A_n)
\end{eqnarray}
$$
so that $m(A_n) \leq \frac{1}{\epsilon^2 n^2}$. Then $\sum A_n < \infty$ and so by Borel-Cantelli, $\limsup A_n = 0$, so for almost all $x$, $\limsup |f_n(x)/n| < \epsilon$. Since $\epsilon$ was arbitrary, we’re done.