 Solution

For the Metropolis–Hastings algorithm, choose a proposal transition $q(x,y)$ and initialize at a state with $\pi (x)>0$. From current $x$, propose $Y\sim q(x,\cdot )$ and accept it with probabilityOtherwise stay at $x$. The accepted off-diagonal transition satisfies detailed balance, since $\pi (x)q(x,y)\alpha (x,y)=\min \{\pi (x)q(x,y),\pi (y)q(y,x)\}$ is symmetric in $x,y$. Thus $\pi$ is stationary. For convergence from arbitrary starting states in the support, choose the proposal so the resulting chain is irreducible and aperiodic. Only ratios of target probabilities are needed, so an unknown normalizing constant cancels.