I was working for my final exam in analysis from Aliprantis and Burkinshaw's Principles of Real Analysis. I got stuck at this problem. Any help is appreciated.
If $\{f_n\}$ is a norm bounded sequence of $L^2[0,1]$, then show that $\dfrac{f_n}{n} \overset{\text{a.e.}}{\longrightarrow}0$.
Let $\|f_{n}\|_{L^{2}}\leq C<\infty$ for each $n$, then \begin{align*} \int\sum_{n}\dfrac{|f_{n}(x)|^{2}}{n^{2}}dx&=\sum_{n}\dfrac{1}{n^{2}}\int|f_{n}(x)|^{2}dx\\ \ \\ &\leq C^{2}\sum_{n}\dfrac{1}{n^{2}}<\infty, \end{align*} so \begin{align*} \sum_{n}\dfrac{|f_{n}(x)|^{2}}{n^{2}}<\infty~~~~\text{a.e.} \end{align*} and hence \begin{align*} \dfrac{f_{n}(x)}{n^{2}}\rightarrow 0 \end{align*} for all such $x$.