Suppose the matrices $A$ and $B$ commute. Do there exists sequences $A_n$ and $B_n$ of matrices such that

1. $A_n \rightarrow A$, $B_n \rightarrow B$.

2. Each $A_n$ is diagonalizable and the same for each $B_n$.

3. For every $n$, $A_n$ commutes with $B_n$.

Moreover, it would be nice if the following property was additionally satisfied: if $A,B$ are real, then $A_n,B_n$ can be chosen to be real as well.

P.S. I asked this question on math.SE a couple of days ago.