 Solution

Let $w(\theta )=\pi (\theta )/\mu (\theta )$ be the importance weight. The two normalized densities giveSince $|F-Z|<C$, there is a finite constant $M$ such that $w(\theta )\le M$ everywhere. The Independence Metropolis–Hastings algorithm has an accepted transition density satisfyingConsequently the whole state space is a small set, with the one-step minorization condition $K(x,\cdot )\ge M^{-1}\pi (\cdot )$. Iterating this Doeblin condition gives uniform geometric convergence in total variation distance, so the chain is geometrically ergodic.