Induction on the integers

Builds on Integers Mathematical induction

Statement

Let $P(z)$ be a property of integers $z$ . Suppose that $P(0)$ holds. Suppose that for each natural number $n$ , $P(n)$ implies $P(n + 1)$ . Suppose that for each natural number $n$ , $P(-n)$ implies $P(-n - 1)$ . Then $P(z)$ holds for every integer $z$ .

Remarks

The proof applies ordinary induction two times: one time to $P(n)$ and one time to $P(-n)$ , for natural numbers $n$ . Each integer is $n$ or $-n$ for some natural number $n$ , thus the two inductions cover all of $\mathbb{Z}$ .

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Tag	`0HBS1F00DN`
URI	`m2://0HBS1F00DN`
Kind	statement, theorem
Writing	W0 stub
MSC	11A99
Status	This copy of the site has no build data. Run `m2 status 0HBS1F00DN`.
Uses	Integers, Mathematical induction

Induction on the integers

Statement

Remarks

Reading path

Further reading

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