  In his first 1958 paper on zonal spherical functions Harish-Chandra proved an
extremely delicate convergence theorem which was basic to his subsequent
definition of his Schwartz space and his theory of cusp forms. This paper gives
elementary proofs that a related integral converges for for groups of real rank
one, several groups of real rank 2 (including $SO(n,2), Sp_4(R)$ and
$Sp_4(C)$), $GL(n,R)$ and $GL(n,C)$. In fact, a stronger result has been proved
in <cite>raphael</cite>. Applications of the question are also studied.
