In the second edition of the Robert & Casella book (Monte Carlo Statistical Methods), the authors have a result, Theorem 7.8, on the independent Metropolis-Hastings sampler: Letting $f$ be the density of the target measure an $g$ the density of the proposal, if a minimization condition is satisfied $f(x)\leq M g(x)$ for all $x$ in the support of $f$, then the chain is uniformly ergodic and $$ \| K^n(x,\cdot) - f\|_{TV} \leq 2 (1-1/M)^n $$
The proof begins with an odd result (equation 7.9), \begin{equation} \|K(x,\cdot) -f\|_{TV} = 2 \sup_{A} |\int_{A} K(x,y) - f(y) dy|. \end{equation} Where is this factor of 2 coming from? Ignoring the factor of 2 for a moment, the authors then obtain \begin{equation} \|K(x,\cdot) -f\|_{TV} \leq 2 (1- 1/M), \end{equation} which I agree with, except, again, for the factor of 2. They then say that, by induction, one can obtain the result.
The best that I can obtain is $$ (2\|K^n(x, \cdot) - f\|_{TV})\leq (2\|K(x, \cdot) - f\|_{TV})^n \leq (2(1-1/M))^n, $$ which is based on a computation in Roberts & Rosenthal (2004) in Probability Surveys.
This calculation has been driving me nuts.