I want to prove that given $A\in \cal{M}_n$ and $\epsilon>0$, there exists a matrix norm $||| \cdot |||$ such that $$||| A|||\leq \rho(A)+\epsilon$$ where $\rho(A)$ si the spectral radius of $A$.
In the proof that I have seen it is shown that $|||A|||= \rho(A)+\epsilon$.
My question (maybe trivial sorry) is: I should prove the relation with $\leq$, so I could simply say that if I take an $\epsilon_1\geq \epsilon$ and then $$||| A|||\leq \rho(A)+\epsilon\leq \rho(A)+\epsilon_1$$ Do you think my idea to pass from $=$ to $\leq$ is right? I think the passage is trivial since it is not specified in the proof, but I can't understand.