$\newcommand{\var}{\operatorname{var}}$
Given $X_1,X_2,\ldots$ is a sequence of random variables with $\rho=0$ and finite second moments.
I am trying to show that
$\var(\bar{X}_n)\rightarrow 0\text{ as }n\rightarrow\infty$
The obvious bit is that $\bar{X}_n=\frac1n\sum_{i=1}^n X_i$ and the exercise allows for the assumption that
$\frac1n\var(X_n)\rightarrow 0\text{ as }n\rightarrow\infty$
Since there is no mention on variance being bounded, how does this change things, if at all?