Let $\displaystyle (X_n)_{n \geq 0}$ be i.i.d random signs, $\displaystyle P(X_n=1)=P(X_{n-1}=-1)=1/2$. Let $\displaystyle (a_n)_{n \geq 0}$ be a sequence of real numbers such that $\displaystyle \Sigma _{n=1} ^{\infty}a_n ^2 < \infty$. Show that $\displaystyle \Sigma _{n=1} ^ \infty a_n X_n $ is almost sure convergent.

By $\displaystyle \Sigma _{n=1} ^{\infty}a_n ^2 < \infty$, $\displaystyle a_n \rightarrow 0$ as $\displaystyle n \rightarrow \infty$. but i dont know how to go from here.