An inequality of Harish-Chandra

In paper I of his masterpiece Harmonic Analysis on Real Reductive Groups, Harish-Chandra included an important inequality that is useful in proving that certain key integrals depending on a parameter converge for large values of the parameter. His proof involved the Tarski-Seidenberg Theorem. The purpose of this note is an elementary proof of the inequality which is an expansion of the idea in my Real Reductive Groups I. This exposition fixes several critical misprints in the original and can be considered to be an erratum for the book.