Let $A$ and $B$ be symmetric, positive definite $N\times N$ matrices. Define $a_n$ as the $n$-th column of $A$ and $a_{n,n}$ as the $n,n$-th element of $A$. I'd like to show that
$$\left[\frac{a_n^T}{a_{n,n}}\left(A-\frac{a_n a_n^T}{a_{n,n}}+B\right)^{-1}\frac{a_n^T}{a_{n,n}}\right]^{-1}\leq b_{n,n}$$
where $b_{n,n}$ is the $n,n$-th element of $B$. (Or alternatively find a case where this inequality doesn't hold.)