  This paper presents an elementary proof of stochastic stability of a
discrete-time reversible Markov chain starting from a Foster-Lyapunov drift
condition. Besides its relative simplicity, there are two salient features of
the proof: (i) it relies entirely on functional-analytic non-probabilistic
arguments; and (ii) it makes explicit the connection between a Foster-Lyapunov
function and Poincar\'e inequality. The proof is used to derive an explicit
bound for the spectral gap. An extension to the non-reversible case is also
presented.
