Prove that if $\displaystyle (g_n)$ is a sequence in $\displaystyle L^2[0, 1]$ with $\displaystyle || g_n ||_2 \leq 1$ for all $\displaystyle n \geq 1$ then $\displaystyle (g_n/n)$ converges to zero a.e.

This should follow from this:

If $\displaystyle (f_n)_n$ is a sequence in $\displaystyle L^1[0, 1]$ is such that $\displaystyle \sum_{n=1}^{\infty} || f_n ||_1 < \infty$ then $\displaystyle \sum_{n=1}^{\infty} | f_n(s) | < \infty$ for almost every $\displaystyle s \in [0, 1]$.

However, I do not see how this would follow. How do I prove this?