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Mathematical induction
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=== Limited mathematical induction === If one wishes to prove that a property {{math|''P''}} holds for all natural numbers less than or equal to a fixed {{mvar|N}}, proving that {{math|''P''}} satisfies the following conditions suffices:<ref>{{Cite book |last=Smullyan |first=Raymond |title=A Beginner's Guide to Mathematical Logic |publisher=Dover |year=2014 |isbn=978-0486492377 |pages=41}}</ref> # {{math|''P''}} holds for 0, # For any natural number {{mvar|x}} less than {{mvar|N}}, if {{math|''P''}} holds for {{mvar|x}}, then {{math|''P''}} holds for {{math|''x'' + 1}}
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