Question: Let $W_n$ be a sequence of independent Bernoulli random variables with $$P(W_n=0)=1-\frac{1}{n^2} \ \ \textit{and} \ \ P(W_n=1)=\frac{1}{n^2}.$$ Let $Y_n$ be a sequence of independent real-valued random variables with $E(Y_n)=0$ and $E(Y_n^2)=n^2$. Assume that $W_n$ and $Y_n$ are independent. Show that the sum $\sum_{i=1}^n W_iY_i$ is almost surely convergent.
I have tried everything with this question. I have come so far to show that $W_iY_i$ converges almost surely to zero. But i have difficulty concluding anything about the sum.
$\sum P(W_n=1) <\infty$. By Borel - Cantelli Lemma it follows that $P(W_n=1 i.o.) =0$. So $(W_n)$ is eventually $0$ with probability $1$ and $\sum W_nY_n$ is convergent for any sequence of random variables $(Y_n)$!