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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 $1$?

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

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