# How to Prove that $cond(A)\ge \frac{||A||}{||C||}$ for any induced matrix norm. With $A$ invertible and $A+C$ singular matrix

How could I prove that for any induced matrix norm

$$cond(A)\ge \frac{\|A\|}{\|C\|}$$

where $$A$$ is an invertible square matrix and $$A+C$$ is a singular matrix?

If $$A$$ is invertible and $$A+C$$ is singular then $$\Vert A^{-1}C\Vert\ge1$$. Because if $$\Vert A^{-1}C\Vert<1$$, then $$I+ A^{-1}C$$ would be invertible and consequently $$A+C=A(I+A^{-1}C)$$ would also be invertible.
Now we have $$\Vert A\Vert\leq \Vert A\Vert \Vert A^{-1}C\Vert\leq \Vert A\Vert \Vert A^{-1}\Vert \Vert C\Vert=\text{cond}(A)\Vert C\Vert.$$ Done.$$\qquad\square$$