
Sequence, Lebesgue Space
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?

Why not apply the quoted result to $\displaystyle f_n=\frac{g_n^2}{n^2}$?... Try to find how to conclude from there.