Non-standard Calculus - Extreme Value Theorem

Extreme Value Theorem

To show that a real continuous function f on has a maximum, let N be an infinite hyperinteger. The interval has a natural hyperreal extension. The function ƒ is also naturally extended to hyperreals between 0 and 1. Consider the partition of the hyperreal interval into N subintervals of equal infinitesimal length 1/N, with partition points x_i = i /N as i "runs" from 0 to N. In the standard setting (when N is finite), a point with the maximal value of ƒ can always be chosen among the N+1 points x_i, by induction. Hence, by the transfer principle, there is a hyperinteger i₀ such that 0 ≤ i₀ ≤ N and for all i = 0, …, N (an alternative explanation is that every hyperfinite set admits a maximum). Consider the real point

where st is the standard part function. An arbitrary real point x lies in a suitable sub-interval of the partition, namely, so that st(x_i) = x. Applying st to the inequality, we obtain . By continuity of ƒ we have

.

Hence ƒ(c) ≥ ƒ(x), for all x, proving c to be a maximum of the real function ƒ. See Keisler (1986, p. 164).