Inverse Transform Sampling for the Exponential Distribution

Let $U \sim \text{Uniform}(0, 1)$.

(a) For a continuous random variable $X$ with strictly increasing CDF $F$, prove that $F(X) \sim \text{Uniform}(0,1)$.

(b) Use part (a) to argue that $X = F^{-1}(U)$ has CDF $F$.

(c) For $X \sim \text{Exp}(\lambda)$ with CDF $F(x) = 1 - e^{-\lambda x}$ ($x \geq 0$), derive $F^{-1}$ explicitly and write a one-line formula for generating exponential samples from uniform samples.