The set of all real or complex invertible matrices is dense I'm trying to show that the set of all invertible matrices $\Omega$ is dense over $F=\mathbb R$ or $\mathbb C$. Let $A\in\Omega$ and $C\in M_{n\times n}(F)$. Since $\|A-C\|<\frac{1}{||A^{-1}||}$, and $\lambda\neq 0$, $A\in\Omega\implies \lambda A\in\Omega$, we have for $\lambda$ large $\|\lambda A-C\|<\frac{1}{\lambda \|A^{-1}\|}$ is small. So for any open ball $B_{\epsilon}(C)$ centered at $C$, we have $\frac{1}{\epsilon\|A^{-1}\|}<|\lambda|\implies \|\lambda A-C\|<\epsilon$ and $\lambda\in\Omega$.
Does this proof look correct? Also, could you use the same idea to prove that the diagonalizable matrices are dense in $M_{n\times n}(F)$, by just letting $A$ be a diagonal matrix with nonzero eigenvalues?

No, it is all wrong.  What you need to show is that given $C \in M_{n \times n}(F)$ and $\epsilon > 0$ there exists $A \in \Omega$ with $\|A - C\| < \epsilon$.  But you are assuming that $A \in \Omega$ with $\|A - C\| < 1/\|A\|$.  Then you make the absurd assertion that $\|\lambda A - C \| < \dfrac{1}{\lambda \|A\|}$.  Actually $\|\lambda A - C\| \ge |\lambda| \|A\| - \|C\| \to +\infty$ as $\lambda \to \infty$, not $0$.
