I wish to get feedback as to whether my answer to the following is correct/adequate.
Attached for your review.
The critical step in the proof will be that the Var(1/n Sigma_k X_k^2) = 1/n^2 * nk = k/n. Since k is fixed, we take the limit as n -> infinity and get the variance to be 0 which means its consistent.
You need to specify this and make it clear otherwise you probably won't get the marks.