Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 43 / 1 / a / ii

For any measurable set $A$, integrating the joint acceptance event in rejection sampling givesSetting $A$ equal to the whole support yields $P(\text{accept})=1/C$. ThusTo verify repeated attempts directly, the probability that the first accepted observation falls in $A$ isWith finite $C$, eventual acceptance has probability one. Applying the same argument after each success proves that the retained sequence consists of independent draws from the target.