# CDF and uniformly distributed random variable

I am trying to solve a basic exercise about random variables, but I am having some trouble.

Let $$Y$$ be an uniformly distributed random variable on $$[0,1]$$ and $$F$$ an arbitrary CDF. Define, for every $$y\in(0,1)$$, $$G(y)=\sup\{x\in\mathbb{R};F(x) and show that the randon variable $$X=G(Y)$$ satisfies $$F_X=F$$.

I am really puzzled with this question. It is very basic, I suppose, but I am still struggling with the concepts, and I think if I solved it, I could unterstand the underlying definitions better.

Here is some of my thoughts: $$F_X(x)=\mathbb{P}(X\le x) = \mathbb{P}(G(Y)\le x).$$

I also know that, since $$Y$$ is uniformly distributed, $$F_Y(x)=x$$. (Is this correct?) But I am having some trouble to connect this to the CDF of $$G(Y)$$. Any hints would be apprreciated.

Let's assume for simplicity that $$Y$$ is a continuous random variable. Then $$G(Y) \leq x$$ iff $$Y \leq F(x)$$ by definition of $$G$$, which happens with probability $$F(x)$$ (over $$Y$$).
When $$Y$$ is an arbitrary random variable, a similar reasoning should work (if the statement is true), but you have to be more careful since $$Y$$ could have atoms.