I was reading Stein's book on Complex Analysis and, at some point, he claims that, for each $\epsilon > 0$, and $0 \leq a_0 \leq 1$, there is a constant $c_{\epsilon}$ such that $$ |\zeta(s)| \leq c_{\epsilon}|b|^{1-a_{0}+\epsilon}, $$ where $\zeta(s)$ is the Riemann Zeta function and $s = a+bi$ is an arbitrary complex number such that $a \geq a_0$ and $|b| \geq 1$. He proves that, under such conditions for $s$,
$$ (\star) \qquad|\zeta(s)| \leq \left|\frac{1}{s-1}\right| + 2|s|^{1-a_{0} + \epsilon} \sum_{n = 0}^{\infty}\frac{1}{n^{\epsilon}}. $$ He then says the claim follows. I understand that the summation is a constant depending on $\epsilon$, but I can't figure out why he can, from $(\star)$, majors $\zeta(s)$ by some expression that only depends on $b$. Can someone help, please?
