TITLE: How to prove that $ {\mathbf{GL}_{n}}(\mathbb{R}) $ is dense in $ {\mathbf{M}_{n}}(\mathbb{R}) $
QUESTION [10 upvotes]: This question is inspired by an old question on here.

Prove:
$ {\mathbf{GL}_{n}}(\mathbb{R}) $ is dense in $ {\mathbf{M}_{n}}(\mathbb{R}) $

Proof:
$ {\mathbf{GL}_{n}}(\mathbb{R}) = \{A \in {\mathbf{M}_{n}}(\mathbb{R}) : \det A \neq 0\}$ where $ {\mathbf{M}_{n}}(\mathbb{R}) $ consists of all $n \times n$ matrices with real entries.
We want to prove that $ {\mathbf{GL}_{n}}(\mathbb{R}) $ is dense in $ {\mathbf{M}_{n}}(\mathbb{R}) $.
$ {\mathbf{GL}_{n}}(\mathbb{R}) $ is dense in $ {\mathbf{M}_{n}}(\mathbb{R}) $ if every matrix $A \in {\mathbf{M}_{n}}(\mathbb{R}) $ either belongs to $ {\mathbf{GL}_{n}}(\mathbb{R}) $ or is a limit point of $ {\mathbf{GL}_{n}}(\mathbb{R}) $.
A matrix $P$ is a limit point of $ {\mathbf{GL}_{n}}(\mathbb{R}) $ in $ {\mathbf{M}_{n}}(\mathbb{R}) $ if and only if every open set $U$ containing $P$ also contains  a point of $ {\mathbf{GL}_{n}}(\mathbb{R}) $ different from
$P$.

And now when we should start to think I am not sure what to do as I have never worked with $ {\mathbf{GL}_{n}}(\mathbb{R}) $ or $ {\mathbf{M}_{n}}(\mathbb{R}) $ before.
I would appreciate a hint or a few on how to continue.

REPLY [4 votes]: Let $A \in M_n(K)$, with $K=\mathbb{R}$ or $\mathbb{C}$, then $\chi_A(\lambda)=\det\left(A-\lambda I_n \right)$ is a nonzero polynomial, so it has a finite number of roots. So it exits a $p \in \mathbb{N}$ such that $\forall k>p \text{, }$ $\chi_A(1/k)\neq 0$ and then the sequence $$\left(A-\frac{1}{k} \cdot I_n\right)_{k>p}\longrightarrow A$$
is made of invertible matrices. 
QED.