  The aim of this note is to present an elementary proof of a variation of
Harris' ergodic theorem of Markov chains. This theorem, dating back to the
fifties essentially states that a Markov chain is uniquely ergodic if it admits
a ``small'' set which is visited infinitely often. This gives an extension of
the ideas of Doeblin to the unbounded state space setting. Often this is
established by finding a Lyapunov function with ``small'' level sets. This
topic has been studied by many authors (cf. Harris, Hasminskii, Nummelin, Meyn
and Tweedie). If the Lyapunov function is strong enough, one has a spectral gap
in a weighted supremum norm (cf. Meyn and Tweedie).
  Traditional proofs of this result rely on the decomposition of the Markov
chain into excursions away from the small set and a careful analysis of the
exponential tail of the length of these excursions. There have been other
variations which have made use of Poisson equations or worked at getting
explicit constants. The present proof is very direct, and relies instead on
introducing a family of equivalent weighted norms indexed by a parameter
$\beta$ and to make an appropriate choice of this parameter that allows to
combine in a very elementary way the two ingredients (existence of a Lyapunov
function and irreducibility) that are crucial in obtaining a spectral gap.
  The original motivation of this proof was the authors' work on spectral gaps
in Wasserstein metrics. The proof presented in this note is a version of our
reasoning in the total variation setting which we used to guide the
calculations in arXiv:math/0602479. While we initially produced it for that
purpose, we hope that it will be of interest in its own right.
