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:


A $\text{Beta}(\alpha,\beta)$ distribution has support on $[0,1]$ with density proportional to $x^{\alpha-1}(1-x)^{\beta-1}$.
You have found that $F_Y(y)=y^{0.5}$ when $0<y<1$
so its density is the derivative $f(y)=0.5y^{-0.5} = 0.5y^{0.5-1}(1-y)^{1-1}$
which is clearly a $\text{Beta}(0.5,1)$ density
