I'm a little confused about a line in the proof of Proposition 1.6 from Stein-Shakarchi's Complex Analysis Chapter 7, Section 1.1 (page 187):
\begin{equation} c |\zeta^{-3} (\sigma)| |{t}|^{- \epsilon} \geq c'(\sigma -1)^3 |t|^{- \epsilon}, \end{equation} where $s = \sigma + i t$, $\sigma \geq 1$, $|t| \geq 1$.
I'm confused about this step because from the definition of the zeta function, $\zeta(\sigma) \geq \int_1^\infty \frac{1}{n^\sigma} d n = \frac{1}{\sigma}$. We have $\zeta(\sigma) \leq 1 + \frac{1}{\sigma}$, but I cannot figure out how to absorb the extra constant $1$ into the factor $c'$.