Let $A \in M_n(\mathbb C)$ and $\lambda$ is an eigenvalue. Does there exist a sequence of diagonalizable matrices $D_n$ and a sequence $\{\lambda_n\}$ complex numbers such that each $\lambda_n$ is an eigenvalue of $D_n$, $D_n \to A$ and $\lambda_n \to \lambda$ ?
I know that the set of diagonalizable matrices are dense in $M_n(\mathbb C)$ , but I'm not sure whether that implies that given a matrix and an eigenvalue of it, whether we can approximate the eigenvalue via eigenvalues of some sequence diagonalizable matrices which approximates the matrix.
Please help.