The The extreme value theorem was stated. Before we can prove it, we need to establish some preliminaries, which turn out to be interesting for their own sake.
Definition: Let C be a subset of the real numbers whose definition can be expressed in the type of language to which the transfer principle applies. Then C is compact if for every hyperreal number x satisfying the definition of C, the standard part of x exists and is a member of C.
To understand the content of this definition, we need to look at the two ways in which a set could fail to satisfy it.
First, suppose U is defined by x≥0. Then there are positive infinite hyperreal numbers that satisfy the definition, and their standard part is not defined, so U is not compact. The reason U is not compact is that it is unbounded.
Second, let V be defined by 0 ≤x<1. Then if dxis a positive infinites-imal, 1 −dx satisfies the definition of V, but its standard part is 1, which is not in V, so Vis not compact. The set Vhas boundary points at 0 and 1, and the reason it is not compact is that it doesn’t contain its right-hand boundary point. A boundary point is a real number which is infinitesimally close to some points inside the set, and also to some other points that are on the outside.
We therefore arrive at the following alternative characterization of the notion of a compact set, whose proof is straightforward.
Theorem: A set is compact if and only if it is bounded and contains all of its boundary points.
Intuitively, the reason compact sets are interesting is that if you’re standing inside a compact set and start taking steps in a certain direction, without ever turning around, you’re guaranteed to approach some point in the set as a limit. (You might step over some gaps that aren’t included in the set.) If the set was unbounded, you could just walk forever at a constant speed. If the set didn’t contain its boundary point, then you could asymptotically approach the boundary, but the goal you were approaching wouldn’t be a member of the set.
The following theorem turns out to be the most difficult part of the discussion.
Theorem: A compact set contains its maximum and minimum.
Proof: Let Cbe a compact set. We know it’s bounded, so let Mbe the set of all real numbers that are greater than any member of C. By the completeness property of the real numbers, there is some real number x between Cand M. Let ∗Cbe the set of hyperreal numbers that satisfies the same definition that Cdoes.
Every real x' greater than x fails to satisfy the condition that defines C, and by the transfer principle the same must be true if x' is any hyperreal, so if dx is a positive infinitesimal, x+ dx must be outside of∗C.
But now consider x−dx. The following statement holds for the reals: there is no number x' <x that is greater than every member of C. By the transfer principle, we find that there is some hyperreal number q in ∗C that is greater than x−dx. But the standard part of q must equal x, for otherwise stq would be a member of Cthat was greater than x. Therefore xis a boundary point of C, and since C is compact, x is a member of C. We conclude Ccontains its maximum. A similar argument shows that C contains its minimum, so the theorem is proved.
There were two subtle things about this proof. The first was that we ended up constructing the set of hyperreals ∗C, which was the hyperreal “big brother” of the real set C. This is exactly the sort of thing that the transfer principle does notguarantee we can do. However, if you look back through the proof, you can see that ∗C is used only as a notational convenience. Rather than talking about whether a certain number was a member of ∗C, we could have referred, more cumbersomely, to whether or not it satisfied the condition that had originally been used to define C. The price we paid for this was a slight loss of generality. There are so many different sets of real numbers that they can’t possibly all have explicit definitions that can be written down on a piece of paper. However, there is very little reason to be interested in studying the properties of a set that we were never able to define in the first place. The other subtlety was that we had to construct the auxiliary point x−dx, but there was not much we could actually say about x−dx itself. In particular, it might or might not have been a member of C.
For example, if C is defined by the condition x= 0, then ∗C likewise contains only the single element 0, and x−dxis not a member of ∗C. But if C is defined by 0 ≤x≤1, then x−dx is a member of ∗C.
The original goal was to prove the extreme value theorem, which is a statement about continuous functions, but so far we haven’t said anything about functions.
Lemma: Let f be a real function defined on a set of points C. Let D be the image of C, i.e., the set of all values f(x) that occur for some xin C. Then if fis continous and C is compact, D is compact as well. In other words, continuous functions take compact sets to compact sets. Proof: Let y= f(x) be any hyperreal output corresponding to a hyperreal input xin ∗C. We need to prove that the standard part of y exists, and is a member of D. Since C is compact, the standard part of x exists and is a member of C. But then by continuity y differs only infinitesimally from f(stx), which is real, so sty= f(stx) is defined and is a member of D.
We are now ready to prove the extreme value theorem, in a version slightly more general than the one originally given on page 56.
The extreme value theorem: Any continuous function on a compact set achieves a maximum and minimum value, and does so at specific points in the set.
Proof: Let f be continuous, and let C be the compact set on which we seek its maximum and minimum. Then the image D as defined in the lemma above is compact. Therefore D contains its maximum and minimum values.
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