Let $\Sigma$ be the population covariance matrix and $\hat{\Sigma}$ be the sample covariance matrix. It is well known that $\hat{\Sigma} \rightarrow \Sigma$ in the large sample limit.
I have also heard that $\hat{\Sigma}^{-1} \rightarrow \Sigma^{-1}$ also holds, but I am not sure how to prove it. Does anyone know how?
To me, this seems similar to arguments involving continuous functions, i.e., if $x_n \rightarrow x$ then $f(x_n) \rightarrow f(x)$.