Let $A$, $B$ be symmetric positive definite matrices. We know
$$ A^{-1}-B^{-1}=-C $$
for positive semidefinite matrix $C$.
I want to show that $A-B$ is positive semidefinite. Here's my take:
\begin{aligned} A^{-1}+-B^{-1}&=-C\\ (A^{-1}+-B^{-1})^{-1}&=-C^{-1}\\ -A(A-B)^{-1}B&=-C^{-1}\\ (A-B)^{-1}&=(BCA)^{-1}\\ A-B&=BCA \end{aligned} where the third line uses the identity $\left(\mathbf{A}^{-1}+\mathbf{B}^{-1}\right)^{-1}=\mathbf{A}(\mathbf{A}+\mathbf{B})^{-1} \mathbf{B}=\mathbf{B}(\mathbf{A}+\mathbf{B})^{-1} \mathbf{A}$.
Now $A$,$B$ are positive definite and $C$ positive semidefinite. Do we know of a result that shows $BCA$ and so $A-B$ is positive semidefinite?
I would appreciate your confirmation. Thanks in advance!