Example
Let be a fixed number, and let be the set of pairs of numbers whose product is at least . When defined over the positive real numbers, has infinitely many minimal elements of the form, one for each positive number ; this set of points forms one of the branches of a hyperbola. The pairs on this hyperbola are minimal, because it is not possible for a different pair that belongs to to be less than or equal to in both of its coordinates. However, Dickson's lemma concerns only tuples of natural numbers, and over the natural numbers there are only finitely many minimal pairs. Every minimal pair of natural numbers has and, for otherwise the pair would belong to and be at least as small in both coordinates, contradicting the assumption that is minimal. Therefore, over the natural numbers, has at most minimal elements, a finite number.
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