Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Question

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$.

My Sol:

$P(Y \leq y ) = P(F(X) \leq y) = P\left(F^{-1}(F(X)) \leq F^{-1}(y)\right) = P(X \leq F^{-1}(y)) = F(F^{-1}(y)) = y$

so at the endpoints we have that $P(Y \leq y) = 1$ for $y \geq 1$ and $P(Y \leq y) = 0$ for $y\leq0$ therefore showing $Y= F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{-1}(Y)$.