Using $\mathrm{maxmag}(A)=\max_{x\neq 0}\frac{\|Ax\|}{\|x\|}$, and $\mathrm{minmag}(A)=\min_{x\neq 0}\frac{\|Ax\|}{\|x\|}$
I found this quite simple to prove using a proposition stating that $\kappa(A)=\frac{\mathrm{maxmag}(A)}{\mathrm{minmag}(A)}$ for all nonsingular A.
However, I think that propostion follows from the conclusion I am trying to prove. I don't think I can use it because I can't prove it without the using relationship I am trying to use it to prove, if that makes sense. How can I use only the definitions of maxmag and minmag to prove this?