# Sequence, Lebesgue Space

• December 13th 2009, 01:48 PM
canberra1454
Sequence, Lebesgue Space
Prove that if $(g_n)$ is a sequence in $L^2[0, 1]$ with $|| g_n ||_2 \leq 1$ for all $n \geq 1$ then $(g_n/n)$ converges to zero a.e.

If $(f_n)_n$ is a sequence in $L^1[0, 1]$ is such that $\sum_{n=1}^{\infty} || f_n ||_1 < \infty$ then $\sum_{n=1}^{\infty} | f_n(s) | < \infty$ for almost every $s \in [0, 1]$.
Why not apply the quoted result to $f_n=\frac{g_n^2}{n^2}$?... Try to find how to conclude from there.