Background information: I am doing a problem from my analytic number theory homework, which is asking me to prove the following:
Suppose $s=\sigma +it$,$0<\sigma\leq 1$, $|t|\geq 2$. Suppose $z=x+iy$,$|y|\leq\frac{|t|}{2}$. For any $\epsilon>0$, prove that there exist $A_\epsilon$ such that $$|\zeta(s+z)|\leq A_\epsilon|t|^{\frac{1}{2}-\sigma-x+\epsilon} \tag{1}$$
For all $x<-\sigma$.
I am familiar with the result that given $C$ and $\delta$ positive, then there exist $A_{C,\delta}$ such that$$|\zeta(z)|\leq A_{C,\delta}|t|^{\frac{1}{2}-\sigma-x}\tag{2}$$ For any $z$ such that $-C\leq x \leq \delta$.
Any hint on how to prove this is appreciated. Thank you for your help in advance.