  A well-known folklore result in the MCMC community is that the
Metropolis-Hastings algorithm mixes quickly for any unimodal target, as long as
the tails are not too heavy. Although we've heard this fact stated many times
in conversation, we are not aware of any quantitative statement of this result
in the literature, and we are not aware of any quick derivation from well-known
results. The present paper patches this small gap in the literature, providing
a generic bound based on the popular "drift-and-minorization" framework of
Rosenthal (1995). Our main contribution is to study two sublevel sets of the
Lyapunov function and use path arguments in order to obtain a sharper general
bound than what can typically be obtained from multistep minorization
arguments.
