I am reading Bernhard Riemann's 1859 paper "On the Number of Primes Less Than a Given Magnitude" in which he derives the Zeta functional equation through a contour integral. I am already familiar with the kind of contour he uses from my previous post, but I am still unclear on one aspect of his derivation.
He says:"If the real part of s is negative, then, instead of being taken in a positive sense around the specified domain, this integral can also be taken in a negative sense around that domain containing all the remaining complex quantities, since the integral taken though values of infinitely large modulus is then infinitely small."
What allows Riemann to do this if $s<0$?
EDIT: This is the contour 
Write $-z = R\cdot e^{i\varphi}$ with $- \pi < \varphi < \pi$ on the outer circular arc. Then
$$\lvert (-z)^{s-1}\rvert = R^{\operatorname{Re} s - 1} \cdot e^{-\varphi \operatorname{Im} s} \leqslant e^{\pi \lvert \operatorname{Im} s\rvert}\cdot R^{\operatorname{Re} s - 1}.$$
Choosing the radii $R$ so that $e^z - 1$ remains bounded away from $0$ on that part of the contour, the standard estimate gives
$$\Biggl\lvert\int_{C_R} \frac{(-z)^{s-1}}{e^z-1}\,dz \Biggr\rvert \leqslant 2\pi R \cdot K \cdot e^{\pi \lvert \operatorname{Im} s\rvert} \cdot R^{\operatorname{Re} s - 1} = 2\pi K e^{\pi\lvert \operatorname{Im} s\rvert} \cdot R^{\operatorname{Re} s}.$$
When $\operatorname{Re} s < 0$, this tends to $0$ for $R \to +\infty$. Of course subject to the constraint that $\lvert e^z-1\rvert \geqslant c$ for some $c > 0$ on the circle. This holds for example when we choose $R_k = (2k-1)\pi$ and let $k\to +\infty$.