> 
> Let $X\_{1}, X\_{2}, \ldots$ be independent random variables uniformly
>  distributed on the interval $[-1, 1]$. Let $a\_1, a\_2, \ldots$ be a
>  sequence of real numbers such that $\sum\_{n=1}^{\infty} a\_n^2$
>  converges. Prove that the series $\sum\_{n=1}^{\infty} a\_n X\_n$
>  converges almost surely.
> 
> 
> 

I'm studying for my exam by doing problems in a book but I am stuck on this one. I am pretty new to almost sure convergence and I guess it makes it difficult even moreso for me that each $X\_i$ is uniform on $[-1, 1]$ rather than just positive numbers like $[0, 1]$. 

I would really appreciate your assistance in helping me answer this question. I tried on this one for almost five hours with no luck. I am familiar with common probability inequalities, like Markov's, Chebyshev's etc.