New proofs of the operator monotony of the square root and the inverse

Let $A,B\in B(H)$. We present among others a simple proof of the widely known result stating that if $0\le A\le B$, then $\sqrt{A}\le \sqrt{B}$. The same idea is used to prove that if $0\le A\le B$ and $A$ is invertible, then $B$ too is invertible and $B^{-1}\le A^{-1}$.