How to prove $Y=X^{2}$ is Beta$\left(0.5,\ 1\right)$ if $X$ is Uniform$(0,\ 1)$ 
I've been reading about uniform distributions but I can't see how $Y=X^{2}$ is Beta$\left(0.5,\ 1\right)$ if $X$ is Uniform$(0,\ 1)$. Is there a way to prove this using the cumulative distribution function method?
My attempt:


Yes, it is incredibly straightforward.  The CDF of $Y$ is $\mbox{Pr} \left[Y \le y \right]$.  Now substitute $Y$ for $X^2$ and square root both sides to obtain $\mbox{Pr} \left[X \le \sqrt{y}\right]$.  The square root operator is a one-to-one transformation for the domain $(0,1)$.  Now the CDF of $X$ is  $\mbox{Pr} \left[X \le x\right] =x$.  Replace $x$ with $\sqrt{y}$ to obtain that the CDF of $Y$ is $\sqrt{y}$.  This is equivalent to the regularized incomplete beta function $I_y(.5,1)$, the CDF of a $Beta(.5,1)$ random variable.
