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If $T$ is the transition matrix of a regular Markov process (so for some $k$ all the entries of $T^{k}$ are positive), we know there is a probability vector $P_{\infty}$ so that if $P_{0}$ is any initial probability vector, then $\lim _{k \rightarrow \infty} T^{k} P_{0}=P_{\infty}$.
Show that the matrix $\lim _{k \rightarrow \infty} T^{k}$ has all of its columns equal to $P_{\infty}$.
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