I got stuck with part of a proof of: 
The steps are all clear to me, until it is said:
"Therefore $f_w$ is the eigenvector of $ A+wB$ associated with the eigenvalue lying within for all small enough $w$".
I cannot carry out the calculations that allow me to prove that $f_w$ is an eigenvector of $A+wB$, nor to prove that there is an associated eigenvalue $\lambda_w$. I attach the specifications of the theorem found on page. 31-32 of Linear Operators and their Spectra (Davies).
Thank you in advance for your attention
I write here, as prosecution of the question. I’d like to understand why holds $\forall \phi\in\mathfrak{B}^{*}$ such that $<f,\phi>=1$. Thanks in advance.