I can prove Gerschgorin's circle Theorem:

If $\displaystyle A \in \textbf{C}^{n\times n} $ , then each eigenvalue of $\displaystyle A $ lies in the union of the disks

$\displaystyle D_{i} = \left\{ \lambda \in \textbf{C}: \left| \lambda -a_{ii} \right| \leq \sum _{j \neq i} \left| a_{ij} \right| \right\} $ .

But I am having trouble proving that, if m disks form a connected domain that is disjoint from the other n-m disks, then there are precisely m eigenvalues of $\displaystyle A $ within this domain.