Prove that $A\ge0, B\ge0$ and $A\ge B$ implies $B^{-1}\ge A^{-1}$

Question

Does anyone know how to prove the following:

Suppose $A$ and $B$ are both positive definite and $A - B$ is positive semi-definite.
Show that $B^{-1} - A^{-1}$ is also positive semi-definite.

I really appreciate any comments!

Answer 1

$A\geq B$ implies that $I-A^{-1/2}BA^{-1/2}\geq 0$ and $A^{-1/2}BA^{-1/2}\sim A^{-1}B\sim B^{1/2}A^{-1}B^{1/2}$ and hence $I-B^{1/2}A^{-1}B^{1/2}\geq 0$ which implies $B^{-1}-A^{-1}\geq 0$.