This result is taken from Roberts and Tweedie, Thm 5.1 (Biometrika 1996): Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms.

The proof works by contradiction. Assume that is geometrically ergodic, then there exists a -small set such that for some constant .

Now, for any , define . We can assume that (since in the assumptions, the esssup is taken wrt ). Then, if , Moreover,

• Using that is a small set, we can easily show that (indeed, if contains two distincts elements for sufficiently small, then showing that is arbitrary small which is not possible since this sum is bounded from below by ).
• This allows to choose with chosen sufficiently close to so that . Then, there exists and such that (which can be easily seen by contradiction).

Now, write for any , which is divergent for sufficiently close to 1.