Induksjonsbevis

First principle of finite induction. Let S be a set of positive integers with the following properties:

(a) The integer 1 belongs to S.
(b) Whenever the integer k is in S, the next integer k+1 must also be in S.

Then S is the set of all positive integers.

Proof. Let T be the set of all positive integers not in S, and assume that T is nonempty. The Well-Ordering Principle tells us that T possesses a least element, which we denote by a. Because 1 is in S, certainly a > 1, and so 0 < a-1 < a. The choice of a as the smalles positive integer in T implies that a-1 is not a member of T, or equivalently that a-1 belongs to S. By hypothesis, S must also contain (a-1) + 1 = a, which contradicts the fact that a lies in T. We conclude that the set T is empty and in wonsequence that S contains all the positive integers.

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