I'm being asked to prove that the MLE of the variance of $X \sim exp(\lambda)$ is consistent, without using that fact that $MLE(\hat{\lambda}_{MSE}, \lambda) \xrightarrow[n\to\infty]{} 0$.
To my understanding, using that fact will say that $\hat{\lambda}_{MSE}$ is consistent, and because our variance estimator is a function of $\hat{\lambda}_{MSE}$, specifically continuous function, then our estimator is consistent as well.
But since I can't use that, my only guess is that I need to go through and prove it by myself.
So I defined $z = Var(X)$, and then
$$
\hat{z}_{MLE} = \hat{Var(X)}_{MLE} = (\hat{\lambda}_{MLE})^{-2}
$$
And now,
$$
MSE(\hat{z}_{MLE}, z) = Bias(\hat{z}_{MLE}, z)^2 + Var(\hat{z}_{MLE})
$$
But what can be done from here? how can I calculate the bias of that estimator?
Note that the MLE of $\lambda$ is $\hat{\lambda} = 1/\bar{X}_n$, thus you can use the WLLN and the continuous mapping theorem for $g(x)=1/x$, that is a.e. continuous transformation, to deduce that $$ \frac{1}{\bar{X}_n} \xrightarrow{p} \frac{1}{\mathbb{E}X} = \lambda. $$