Editing Mathematical induction (section)

=== Infinite descent ===
{{main|Infinite descent}}
The method of infinite descent is a variation of mathematical induction which was used by [[Pierre de Fermat]]. It is used to show that some statement {{math|''Q''(''n'')}} is false for all natural numbers {{mvar|n}}. Its traditional form consists of showing that if {{math|''Q''(''n'')}} is true for some natural number {{mvar|n}}, it also holds for some strictly smaller natural number {{mvar|m}}. Because there are no infinite decreasing sequences of natural numbers, this situation would be impossible, thereby showing ([[proof by contradiction|by contradiction]]) that {{math|''Q''(''n'')}} cannot be true for any {{mvar|n}}.

The validity of this method can be verified from the usual principle of mathematical induction. Using mathematical induction on the statement {{math|''P''(''n'')}} defined as "{{math|''Q''(''m'')}} is false for all natural numbers {{mvar|m}} less than or equal to {{mvar|n}}", it follows that {{math|''P''(''n'')}} holds for all {{mvar|n}}, which means that {{math|''Q''(''n'')}} is false for every natural number {{mvar|n}}.