Suppose that $X_1, X_2, X_3,\ldots$ are sequence of independent random variables such that $\mu_k= 0$ and $ \sigma^2_k =\operatorname{Var}(X_k)< \infty$ for all $k$. Then show that $\sum_{k=1}^{\infty} \sigma^2_k <\infty$ implies $|\sum_{k=1}^{\infty} X_k|<\infty $ almost surely.
I was wondering how could I use the facts: 1) if a martingale $M$ is bounded in $L^2 $ then $\lim M_n$ exists almost surely. 2) Orthogonality of increments of $M$ to prove the above statement. I would like to see the solution in explained way.