# Inverse Sampling Method

I want to prove from first principles that if $X$ is a continuous random variable with cdf $F$, then $Y = F(X)$ is uniformly distributed on $[0,1]$.

For $y \in (0,1)$, if we define $a = \sup\{ x: F(x) \leq y \}$, then I know that $$\mathbb{P}(Y \leq y) = \mathbb{P}(X \leq a) = F(a).$$ Using continuity, it's easy to argue that $$F( a ) \leq y ,$$ but I'm having a hard time showing that $F( a ) \geq y$.

But it's really only the boundary case that's confusing me. I am actually able to show that $F(a)$ is ${strictly}$ greater than $y$, but that doesn't seem to be what is needed...

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