I'm having trouble with the following problem:
Let $A\in M_n(\mathbb{C})$ be a Hermitian matrix, $\mu\in\mathbb{C}$, $\epsilon>0$ and $x\in\mathbb{C}^n$ such that $\mid\mid x\mid\mid=1$ and $\mid\mid Ax-\mu x\mid\mid<\epsilon $, then $A$ has an eigenvalue $\lambda$ such that $\mid\lambda-\mu\mid<\epsilon$.
I would like any hints of where to start. I was thinking of using Courant-Fischer, or some of the "spectral inequalities", but I don't know how to proceed.
Thank you.