Let $\{X\_n\}$ be a sequence of uncorrelated random variables with common mean $\mu$, such that $\sup\_n$Var$(X\_n)<\infty$. If $S\_n=\sum\_{k=1}^n X\_k$, show that
$n^{-2}\sum\_{k=1}^nS\_k$ converges in probability as $n\to\infty$ and identify the limit.

I know that using Weak Laws of Large Numbers, it is easily proven that $\frac{S\_n}{n}\to\mu$ in $L^2$, and thus converges to $\mu$ in probability as well. But I'm having trouble using this result to prove that $n^{-2}\sum\_{k=1}^nS\_k$ converges in probability. I think it would help to know the limit (at least intuitively) to prove this, but I'm a bit stuck on this problem.

**My attempt:**
First note that $$\frac{\sum\_{k=1}^nS\_k}{n^2}\leq\frac{nS\_n}{n^2}=\frac{S\_n}{n}$$
Then since we know $E(\lvert \frac{S\_n}{n}-\mu\rvert^2)\to 0$, $E(\lvert\frac{\sum\_{k=1}^nS\_k}{n^2}-\mu \rvert^2)\leq E(\lvert \frac{S\_n}{n}-\mu\rvert^2)$ by monotonicity (since expectations are integrals, and monotonicity holds for integals), and thus $E(\lvert\frac{\sum\_{k=1}^nS\_k}{n^2}-\mu \rvert^2)\to 0$ as $n\to\infty$. Now using Markov's inequality, I can conclude that $\frac{\sum\_{k=1}^nS\_k}{n^2}\to\mu$ in probability.

I kind of get the feeling that my answer may be incorrect. I would love for someone to provide some hints/point me in the right direction. Thanks