Theorem (Law of Large Numbers, non-IID): Suppose $\x_1, \x_2, \ldots$ are independent random variables with $\tfrac{1}{n}\sum_i \bmu_i \to \bmu$ and $\tfrac{1}{n}\sum\Var\x_i$ is bounded. Then $\bar{\x} \inP \bmu$.
Proof. We have
\[\Ex \bar{\x} = \frac{1}{n} \sum_i \bmu_i \to \bmu\]and
\[\Var \bar{\x} = \frac{1}{n^2} \sum_i \Var \x_i\]which goes to zero because $\tfrac{1}{n}\sum\Var\x_i$ is bounded. Therefore $\bar{\x} \inP \bmu$.