Consider $X_1, ..., X_n,...$ independent random variables that satisfy: $E[X_i]=1+\frac{1}{1+i^2}$ and $Var[X_i]=\sqrt{i}$. Show that the chance of $\frac{1}{n}\sum\limits_{i=1}^n X_i$ converges to $1$ as $n \rightarrow \infty$.
My question is whether I can apply the strong law of large numbers, as far as I understand that law it requires $E[X_i]=\mu \neq \mu_i$ Idem for $Var[X_i]$. Could anyone give a hint here?