Non-homogeneous Markov chains with a finite state space and a Doeblin type theorem

Doeblin [1] considered some classes of finite state nonhomogeneous Markov chains and studied their asymptotic behavior. Later Cohn [2] considered another class of such Markov chains (not covered earlier) and obtained Doeblin type results. Though this paper does not present the best possible results, the method of proof will be of interest to the reader. It is elementary and based on Hajnal's results on products of nonnegative matrices