The question I'm working on says:
Let $X_1, X_2, \cdots$ be iid random variables each with mean $\mu$ and variance $\sigma^2$.
a) Determine $$ \lim\limits_{n \to \infty} \frac{X_1^2 + \cdots + X_n^2}{n}$$
I think that the answer involves the strong law of large numbers which in my text book says that: $$ \frac{X_1 + \cdots + X_n}{n} \rightarrow \mu \text{ as } n \rightarrow \infty $$
Intuitively it seems that the law of large numbers is saying that the average of random variebles approachs their average as the number approaches infinity. It seems like it makes sense then that the average of each those random numbers squared as the number approaches infinity is their average squared.
So from that it seem like we have $$ \lim\limits_{n \to \infty} \frac{X_1^2 + \cdots + X_n^2}{n} = \mu^2$$
but if we squared the both sides of the law of large numbers we get $$ \lim\limits_{n \to \infty} \frac{X_1 + \cdots + X_n^2}{n^2} = \mu^2 $$ instead. I think I might be forgetting something from analysis as far as limits of sums. Any help would be appreciated.
