# Matrix norms - question about the inequality $||A||<\rho(A)+\epsilon$

Since for any $$\epsilon>0$$ there exists a matrix norm $$||\cdot||$$ such that $$||A||<\rho(A)+\epsilon$$ holds, why does that imply that there exists a matrix norm $$||\cdot||$$ such that $$||A||<1$$ if the spectral radius $$\rho(A)<1$$ ?

Is this because if $$\rho(A)<1$$ and if $$||A||<\rho(A)+\epsilon$$ has to hold for any $$\epsilon>0$$, however small, then however close $$\rho(A)$$ is to $$1$$, if we add an infinitesimally small $$\epsilon$$ the sum $$\rho(A)+\epsilon$$ will still be less than $$1$$ since there are infinitely many numbers between $$\rho(A)$$ and $$1$$?

• Yes. In particular, let $\epsilon = \dfrac{1}{2}(1-\rho(A))$ – BaronVT Dec 11 '18 at 22:54