For every matrix $A\in\Ma(n,\C)$ and every $r > 0$ there is a matrix $B=(b_{jk})\in\Ma(n,\C)$ such that $\sum|b_{jk}|^2 < r$ and $A+B$ is diagonalizable, i.e. diagonalizable matrices are dense.

We consider $A$ as a linear map $A\in\Hom(\R^n)$! Choosing another basis $b_1,\ldots,b_n$, we may assume w.l.o.g. that the matrix of $A$ is given in its normal form with block diagonal entries of the form $$ J(\l)=\left(\begin{array}{cccccc} \l&1&0&\cdots&0&0\\ 0&\l&1&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\cdots&1&0\\ 0&0&0&\cdots&\l&1\\ 0&0&0&\cdots&0&\l \end{array}\right)\in\Ma(m,\C) $$ Now choose pairwise distinct small numbers $\e_1,\ldots,\e_m$ and define $$ I(\l)=\left(\begin{array}{cccccc} \l+\e_1&1&0&\cdots&0&0\\ 0&\l+\e_2&1&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\cdots&1&0\\ 0&0&0&\cdots&\l+\e_{m-1}&1\\ 0&0&0&\cdots&0&\l+\e_m \end{array}\right)\in\Ma(m,\C) $$ Then $I(\l)$ is diagonalizable because it has $m$ pairwise distinct eigen-values. Replacing the block matrices $J(\l)$ with the corresponding blocks $I(\l)$ we get another matrix, which is close to the matrix of $A$ in the basis $b_1,\ldots,b_n$ and hence the matrix $B$ of this mapping with respect to the canonical basis is close to $A$.