In one sentence
The well-ordering principle: every non-empty subset of the natural numbers has a smallest element; a finite non-empty set of real numbers has both a smallest and a largest element.
The fact is unremarkable, but its use is clever: pick out the “most” object from a collection (the largest, smallest, farthest, the one with the most…), and its extremeness is often exactly the missing piece of the argument. First, let's see how it works in a tournament.
Predict first
players play a round robin (every two play once, no draws). Could the player with the most wins have won only games?
There are games in total, each producing one win, so the average is wins per player. Let each player beat exactly “the next two” ( beats ; beats ; …; beats ): all five have wins.
Stop 1 · Tournaments: the player with the most wins
Theorem: in a round robin, the player with the most wins either beat any other player directly, or beat someone who beat .
The player with the most wins reaches everyone in one or two steps
Step 1 / 4- Grab the extreme
has the most wins () and beat , and directly: the solid lines.
The argument never used how many players there are or who beat whom. It used only one sentence: nobody won more than .
Stop 2 · Sets of numbers: the largest number
Can a finite collection of distinct real numbers be such that each one is the average of two others?
Take the largest number . If with two of the other numbers, then and are both less than , so their average is less than : a contradiction. So no.
Break it
Counterexample lab: remove one condition
Conclusion: The numbers cannot each be the average of two others ← still true?
When the collection is finite, nobody is larger than the largest number, so it cannot be the average of two others.
Drop “finite”: in the set of all integers , every number is the average of and . has no largest number, the extremal principle has nothing to grab, and the conclusion fails.
Stop 3 · Geometry: the closest point and line
The Sylvester–Gallai theorem: if finitely many points in the plane are not all on one line, then some line passes through exactly two of them. (Below is Kelly's proof by the extremal principle.)
Kelly's proof
Step 1 / 4- Pick the extreme
Consider all combinations of a line through at least two of the points and a point not on . There are finitely many such combinations, and at least one (the points are not all collinear). Pick the combination where the distance from to is smallest.
At the three stops, the extreme objects were the player with the most wins, the largest number and the closest point–line pair. The common steps: check that such an object exists (finiteness), pick it, then show that “if the conclusion were false, we could find an object even more extreme”.
Practice
players play a round robin (no draws). At least how many games did the player with the most wins win?
positive integers add up to . At least how large is the largest of them?
real numbers are arranged in a circle, and each equals the average of its two neighbors. Prove that all numbers are equal.
There are points in the plane, with all pairwise distances different. Each point shoots an arrow at the point nearest to it. Prove that at least one point is not hit.