Prove that $\operatorname{cond}(f ) = \|A^{−1}\|\|A\|.$

Question:

I am trying to prove that the condition number of a function $ f(x) = Ax $ is given by:

$$\text{cond}(f) = \|A^{-1}\|\|A\|$$

The definition of the condition number of a function ( f ) is:

$$ \limsup_{\|\Delta x\| \to 0} \frac{\frac{\|\Delta y\|}{\|y\|}}{\frac{\|\Delta x\|}{\|x\|}} $$

Starting from this definition, I have:

$$ \text{cond}(f) = \limsup_{\|\Delta x\| \to 0} \frac{\frac{\|A\Delta x\|}{\|Ax\|}}{\frac{\|\Delta x\|}{\|x\|}} $$

Then, I proceed with the following simplifications:

$$ \limsup_{\|\Delta x\| \to 0} \frac{\|A\|\|\Delta x\|}{\|A\|\|x\|}$$

Next, eliminating ( x ), I arrive at:

$$ \limsup_{\|\Delta x\| \to 0} \frac{\|A\|}{\|A\|}$$

Finally, I get:

$$ \text{cond}(f) = \|A^{-1}\|\|A\|$$However, I feel like something is wrong, but I’m not sure what the exact issue is. Could anyone help me identify where my reasoning or steps might be incorrect?