= Solution
The normalized sum has mean one and variance $(5n-2)/n^2$. For every fixed $\epsilon>0$, the <Chebyshev inequality> gives
$$
0\leq\mathbb P\left(\frac{S_n}{n}>1+\epsilon\right)
\leq\mathbb P\left(\left|\frac{S_n}{n}-1\right|>\epsilon\right)
\leq\frac{5n-2}{n^2\epsilon^2}\longrightarrow0.
$$
This proves the requested limit, and in fact proves <convergence in probability> of $S_n/n$ to one. It is a <finite covariance range weak law>: adjacent dependence is harmless here because only linearly many covariance terms survive.
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