The Extremal Principle
Consider the largest, the smallest, the closest.
Pick the extreme object — the longest edge, the smallest counterexample, the point closest to a line — and derive a contradiction or a forced structure. Because the extreme exists (finiteness or well-ordering guarantees it), the argument is airtight.
In combinatorics it drives most 'show some configuration must occur' proofs: take a vertex of maximum degree, a longest path, a triangle of least area.
Zeitz calls this one of the handful of genuinely universal problem-solving moves, and it pairs naturally with infinite descent.
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