In this expository paper, we prove the following theorem, which may be of some use in studying Markov Chain Monte Carlo methods like hit and run, the Metropolis algorithm, or the Gibbs sampler. Suppose a discrete-time Markov chain is aperiodic, irreducible, and there is a stationary probability distribution. Then from almost all starting points the distribution of the chain at time n converges in norm to the stationary distribution. This known theorem is a special case of more general results due to Doeblin, and the paper concludes with a brief review of the literature.