I'm looking over a sample test question and its answer but I don't see how the answer given actually answers the question.
Here it is verbatim: Prove that the sample standard variance $s^2=\frac{\sum_{i=1}^n(X_i-\bar X)}{n-1}$ is a consistent estimator for the population variance (use n=2).
First of all I'm confused that it says use n=2 since my understanding of consistency is to show that as n goes up the estimator gets closer to the population true variable.
The solution opens with:
Need to show that if $X_1$ and $X_2$ are i.i.d. then
$E[(X_1-\frac{X_1+X_2}{2})^2 + (X_2 - \frac{X_1+X_2}{2})^2] = \sigma^2$
Isn't this what you need to show for unbiasedness not consistency? (notwithstanding that, here's the rest)
$$ \begin{align} E[\underbrace{(X_1-\frac{X_1+X_2}{2})^2 + (X_2 - \frac{X_1+X_2}{2})^2}_{s^2 \mathrm{\ when\ n=2}]} = \sigma^2\tag1\\ \frac{1}{4}E(X_1-X_2)^2 + \frac{1}{4}E(X_2-X_1)^2 \tag2\\ \frac{1}{4}*2E(X_1-X_2)^2\tag3\\ \frac{1}{2}E[(X_1-\mu)-(X_2-\mu)]^2\tag4\\ \frac{1}{2}E(X_1-\mu)^2+E(X_2-\mu)^2\underbrace{+}_{\text{should be minus, no?}}2\underbrace{E(X_1-\mu)(X_2-\mu)}_{\text{becomes 0 because $X_1$ and $X_2$ are iid}}\tag5\\ \frac{1}{2}(\sigma^2+\sigma^2)\tag6\\ \sigma^2\tag7\\ \end{align} $$
(BTW, sorry for the poor alignment, the underbraces don't get along with the &)
So anyway, back to my earlier question, isn't this the proof of unbiasedness not consistency? If this does prove consistency then how when n is constant, what am I missing?