Wikipedia gives a derivation of the Kramers-Kronig relation where they consider an integral of the form $\oint\frac{\chi(\omega^\prime)}{\omega^\prime-\omega}d\omega^\prime$ with $\chi(\omega^\prime)$ assumed to be analytic in the complex upper half plane. Then they argue that
The length of the semicircular segment increases proportionally to $|\omega^\prime|$, but the integral over it vanishes in the limit because $\chi(\omega^\prime)$ vanishes at least as fast as $1/|\omega^\prime|$.
How do they arrive at the conclusion that $\chi(\omega^\prime)$ vanishes at least as fast as $1/|\omega^\prime|$?