Density of diagonalizable complex matrices ($\overline{\{A:A\text{ is diagonalizable}\}}=M_n(\mathbb{C})$)

Perturb each diagonal position of a Jordan normal form matrix by a different arbitrarily small scalar, avoiding the finitely many collisions between diagonal entries. The resulting upper triangular matrix has distinct eigenvalues, hence is diagonalizable. Conjugate back to approximate any complex matrix. Continuous identities proved for diagonalizable matrices can therefore extend to all matrices.