Let $\displaystyle X_1, X_2, ... $ be a sequence of independent random variables with $\displaystyle E(X_i) = \mu_i$ and $\displaystyle Var(X_i) = (\sigma_i)^2$. Show that if $\displaystyle n^{-1} \sum_{i=1}^{n} \mu_i \rightarrow \mu$ and $\displaystyle n^{-2}\sum_{i=1}^{n}{\sigma_i}^2\rightarrow 0$, then $\displaystyle \bar{X} \rightarrow \mu$ in probability.