What is the purpose of envelope function?
Say, for single random variable, why can't we just sample x coordinate and y coordinate (representing probability) and reject all those y's which lay above the curve?
How do you propose sampling $y$? You take a sample from some other distribution $q$. What happens if $q(x) < p(x)$ where p is the desired distribution? You cannot have an acceptance ratio greater than one and so the samples would not be distributed according to $p$.
Thus you need a function $q$ where $q$ is greater than $p$ for all $x$ but is close to $p$ so as to minimise rejection. This might then resemble an envelope.