Solution
= Solution
The identity
$$
\frac1n\sum_{i=1}^n(X_i-\bar X_n)^2
=\frac1n\sum_{i=1}^nX_i^2-\bar X_n^2
$$
and the weak law applied to $X_i$ and $X_i^2$ give convergence in probability to
$$
\boxed{\mathbb EX_1^2-\mu^2=\sigma^2.}
$$
= Solution
The identity
$$
\frac1n\sum_{i=1}^n(X_i-\bar X_n)^2
=\frac1n\sum_{i=1}^nX_i^2-\bar X_n^2
$$
and the weak law applied to $X_i$ and $X_i^2$ give convergence in probability to
$$
\boxed{\mathbb EX_1^2-\mu^2=\sigma^2.}
$$