Weighted Sum of Independent Random Variables

Question

I am trying to prove the following:

Let $\xi_{1},\xi_{2},...$ be independent random variables uniformly distributed on the interval [−1, 1]. Let $a_{1},a_{2},...$ be a sequence of real numbers such that $\sum_{n=1}^{\infty} a_{n}^{2}$ converges. Prove that the series $\sum_{n=1}^{\infty}a_{n}\xi_{n}$ converges almost surely.

Answer 1

Show $E (\sum_n a_n \xi_n)^2 = E \xi_1^2 \sum_n a_n^2$.

Then $(\sum_n a_n \xi_n)^2$ is finite ae.