I am trying to show that the estimator $\frac{n(k-1)S}{(nk-2)}$ is an inconsistent estimator of $\sigma^2$. I must show that the estimator will converge in probability to $\frac{(k-1)\sigma^2}{k}$.
Note that $S = \sum_{i=1}^{n} \sum_{j=1}^{k}$ $\frac{(X_{ij} - \overline{X}_{i})^{2}}{(n(k-1))}$ where $X_{i1},...,X_{ik}$ are iid $\mathcal N(\theta_{i}, \sigma^{2})$.
Edit: For the sake of context, I should note that this is part of a larger problem, and I have reduced the problem to this.