PROOF. Letf.-T*f and gn->g in C, and let an and I3n be the least numbers such that acxf. > gn and f.g1,>f, These exist by Lemma 1 and are positive since S is Archimedean, and 0(fn,, gn) = On satisfies ee9 =ana4n. Let 0= lim inf On. The case 0 =oo is trivial, since it imposes no restriction on O(f, g). Moreover, by restricting attention to a subsequence, we can reduce to the case 0 = lim On. This may increase the values of a = lim inf a. and =lim inf i3n. But both a > O and g > O since, by Lemma 1, af _ g > O and g >f where a 9 ? eo and S is Archimedean. It follows that a 9. Moreover, by Lemma 1, af> g and (e0/a)g >f, whence