If we let $\displaystyle X_1, X_2, \cdots, X_n$ be independent and identically distributed observations from a population with mean $\displaystyle \mu$ and variance $\displaystyle \sigma^2$ then the weak law of large number states that $\displaystyle \bar{X_n} \rightarrow^p \mu $ and I can prove this part, however does $\displaystyle S^2 \rightarrow^p \sigma^2$? Where $\displaystyle S^2 = \frac{1}{n-1} \sum (X_i - \bar{X})^2$ the sample variance? If so, how to prove it?