I have no idea about this problem:
Let $\mathcal{A}$ be a $C^*$-algebra and $x,y$ are self-adjoint elements. Show that if $0 \leq x \leq y$, then $x^{1 / 2} \leq y^{1 / 2}$, where $x^{1/2}$ denotes the unique square root of $x$.
where the partial order $\leq$ is defined as below: $$x \leq y \iff y-x \text{ is a positive element}$$ and a positive element $a$ is an element that can be expressed as $a = b^* b$ for some $b \in \mathcal{A}$.
I've learned the basic concepts about $C^*$-algebras and the continuous functional calculus, but I have no idea how to apply the theory here.
Any help is appreciated.
