Suppose a nonnegative sequence $h=h_n$ satisfying $h\rightarrow 0$ and $nh\rightarrow \infty$ as $n \rightarrow \infty$. Consider the asymptotic notation 'Big Oh' and 'small oh'. Define
\begin{equation} A= \begin{bmatrix} 1+O(n^{-1})&O(n^{-1})\\O(n^{-1})&h^2u+O(n^{-1}) \end{bmatrix}, B= \begin{bmatrix} h^2u+O(n^{-1})\\O(n^{-1}) \end{bmatrix} \end{equation} where $u$ is a nonnegative constant.
I am stucked to show the following result
$$e_1^tA^{-1}B=h^2u+O(n^{-1})+o(h^2),$$ where $e_1=(1,0)^t$.
This problem seems to be straightforward, but it involves many asymptotic operations. I attempted to get the asymptotic terms in many ways but never succeeded.
Can you derive this result or give me some hints?
Thank you in advance.