Probability Integral Transform (Inverse CDF Method)

Let $F_X$ be a continuous, strictly increasing CDF and $U \sim \operatorname{Uniform}(0,1)$. Prove that $Y = F_X^{-1}(U)$ has CDF $F_X$.

Conversely, show that if $X$ has CDF $F_X$, then $F_X(X) \sim \operatorname{Uniform}(0,1)$.