= Solution
Write $Y_n=n^{-1}\sum_{i=1}^n|X_i|$. Truncate each $|X_i|$ at a level $K$. The average of the truncated parts is at most $K$, while the expected average of the tails is
$$
\mathbb E[|X_1|\mathbf1_{|X_1|>K}],
$$
independently of $n$. Markov's inequality then shows that $(Y_n)$ is uniformly integrable as $K\to\infty$. Since $|S_n|/n\leq Y_n$, the <sequence> $(S_n/n)$ is uniformly integrable.
Convergence in probability to the constant $\mathbb EX_1$, together with uniform integrability, implies convergence of first absolute moments by the Vitali convergence theorem. Hence
$$
\boxed{\mathbb E\left|\frac{S_n}{n}-\mathbb EX_1\right|\to0.}
$$
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