I was reading the proof in Bratteli-Robinson's book on $C^{\star}$-algebra that $$\rho(A)=\lim_{n\to \infty}\|A^n\|^{1/n}$$ for some element $A$ of a $C^{\star}$-algebra $\mathcal{U}$ (Prop 2.2.2 page 26).
At some point they set $$r_A=\limsup_{n\to \infty}\|A^n\|^{1/n}$$ and act by contradiction assuming that $\rho(A)<r_A$. At the end, they obtain that \begin{equation}\label{eq}\lim_{n\to\infty}\frac{\|A^n\|}{r_A^n}=0\end{equation} and say this contradict the definition of $r_A$. I do not agree with that as it is easy to construct sequences $v_n$ such that $\lim_{n\to\infty}v_n^{1/n}=a$ and $\lim_{n\to\infty}\frac{v_n}{a^n}=0$. What am I missing ?
Thanks by advance