I'm stuck trying to prove a step inside a lemma from Serre; given is

0<a<b

0<x

To prove:

$\displaystyle |\int_{a}^{b}e^{-tx}e^{-tiy}dt|\leq\int_{a}^{b}e^{-tx}dt$

I've tried using Cauchy-Schwartz for integrals, but this step is too big (using Mathematica, it lead to a contradiction); something simpler must do the trick.

Thanks in advance.