Let \(\left\{X_{n}ight\}_{n=1}\) be a set of independent and identically distributed random variables from a distribution with me \(\mu\) and finite variance \(\sigma^{2}\). Show that

\[S^{2}=n^{-1} \sum_{k=1}^{n}\left(X_{k}-\bar{X}ight)^{2}=n^{-1} \sum_{k=1}^{n}\left(X_{k}-\muight)^{2}+R\]

where \(R \xrightarrow{p} 0\) as \(n ightarrow \infty\).