 Solution

Assume the target probability density function is supported where the proposal probability density function is positive and that the envelope constant is finite. Any $C\ge M$ satisfies $f(x)\le Cg(x)$, and integration gives $C\ge 1$. For each attempt draw independently $Y\sim g$ and $U\sim U(0,1)$. Accept $Y$ if $U\le f(Y)/(Cg(Y))$; otherwise discard it and repeat. Independent attempts give independent accepted values. A finite pre-generated list of independent proposal draws can be screened with independent uniforms in the same way, though the number retained is random. If $M=\infty$, this fixed-envelope rejection sampling method cannot be used with that proposal.