Separability of $M_n(\mathbb{R})$

Countable dense subset. Let $M_n(\mathbb{Q})$ be the set of matrices with rational entries. It is countable (finite product of countable sets).

Density. For any $A\in M_n(\mathbb{R})$ and $\varepsilon>0$, approximate each entry $a_{ij}$ by a rational $q_{ij}$ with $|a_{ij}-q_{ij}|<\varepsilon$. The matrix $Q=(q_{ij})$ satisfies $\|A-Q\|_\infty<\varepsilon$.

Conclusion. $M_n(\mathbb{Q})$ is countable and dense in $M_n(\mathbb{R})$, which is separable.