Given X1, X2, ... an infinite sequence of i.i.d. random variables with finite expected value E(X1) = E(X2) = ... = µ < ∞, we are interested in the convergence of the sample average

This proof uses the assumption of finite variance $\operatorname {Var} (X_{i})=\sigma ^{2}$ (for all $i$). The independence of the random variables implies no correlation between them, and we have that

As n approaches infinity, the expression approaches 1. And by definition of convergence in probability (see Convergence of random variables), we have obtained