Originally Posted by

**eskimo343** Show that if $\displaystyle || f_n ||_1 \leq 2^{-n}$ for every $\displaystyle n \geq 1$ then $\displaystyle (f_n)_n$ converges to zero a.e.

This should follow from

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 don't see how to show this. I would appreciate some hints on how to proceed.