Let be a sample variance from a sample of size from distribution.
Previous exercise gives:
Also is unbiased estimator for , by definition this means:
We need to prove consistency of for , that is, for all :
As written above for all , is a variable with finite variance and mean so we can apply (Theorem 5.7.1.) for all :
Now substitute known mean and variance:
Since this inequality holds for all and limes is monotonous, from (1) follows:
And since limes on the left is a limes of probabilities, it is always smaller than 1.
Thus we have:
So by definition is consistent for