I want to show that:
$$\hat{\sigma^2}=(1/n)\sum^{n}_{i=1} ( X_i-\bar{X} )^2$$
is a consistent estimator of $\sigma^2$.
I was using the Weak Law of Large Numbers in the sense that: $$E(X_i-\bar{X })^2=\sigma^2[1-1/n]$$, and I also showed that $$\operatorname{Var}(X_i-\bar{X })^2< \infty$$
Given those two, I said that the average of $(X_i-\bar{X })^2$ was $\hat{\sigma^2}$, and that the estimator converges in probability to the sigma square.
The problem is that the expectation shows that my estimator is biased (the conditions state that $EX_i$ should be unbiased and equal to $\mu$).
I know the traditional answer to show this is consistent is to plug plus and minus $\mu$ inside my estimator and show that $\bar{X}\to\mathrm{Prob}( \mu_x)$, and also show that it converges in probability to sigma square, then everything is fine.
I just want to know if the first solution is valid or not, or if the expectation of what I consider"Xi" has to be unbiased for $\mu$. Thanks in advance.