We will determine for any monotonically increasing function . Since g is monotontically increasing, . We assume that . We further assume that , so we may take . A simple proof by contradiction, which follows, shows that is an upper bound for g:
Assume that there is a point such that . Let , so . Furthermore, and imply that .
The two assumptions made do not overly restrict the applicability of the proof. If , consider in place of g. If , consider in place of g, such that g' is monotonically increasing. If exists, it may be taken for y; otherwise, a trivial upper bound may be used.
|Jeff Tupper||March 1996|