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

Question

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?

Answer 1

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.$\qquad\square$