Consider distribution function $f(x)$, plotted in the following figure with red color, as a stationary distribution for metropolis algorithm. If we use function $g(x)$ to generate the next state, where $c<g(x)$, or mathematically: $$x'=g(x),\quad \forall x, c<g(x)$$
My argument is that, $c$ acts as a barrier to sampling. And when the algorithm tries to sample from distribution $f$, the minimum value that can be obtained is $c$. As a result, eventually $x=c$ with probability $1$. In the following figure the location of $c$ is specified with blue color.
My question is, with enough iterations, is it possible to prove this claim that eventually $x$ is equal to $c$ with probability $1$?
