= Solution
Write $\bar X_n=n^{-1}\sum_{j=1}^nX_j$ and apply the stated second-moment <Markov inequality> to $Y=\bar X_n-\mu$. The preceding calculation gives $\mathbb E Y^2=\sigma^2/n$. Thus for every $\varepsilon>0$,
$$
\boxed{\mathbb P(|\bar X_n-\mu|>\varepsilon)\leq\frac{\sigma^2}{n\varepsilon^2}\longrightarrow0.}
$$
This is <convergence in probability> to the common mean, proving the finite-variance <weak law of large numbers>. It also covers $\sigma^2=0$, when the bound is identically zero.
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