## Mathematical Induction (Proof)

Posted by guptaradhesh on October 3, 2010

### Mathematical Induction

having had an arbit wave of recalling the proof of mathematical induction, I found it a good part-time read/ recall. Here is the same:

Proof: Take S to be the set of all natural numbers for which P(*n*) is false. Let us see what happens if we assert that S is nonempty. Well-ordering tells us that S has a least element, say *t*. Moreover, since P(0) is true, *t* is not 0. Since every natural number is either zero or some *n*+1, there is some natural number *n* such that *n*+1=*t*. Now *n* is less than *t*, and *t* is the least element of S. It follows that *n* is not in S, and so P(*n*) is true. This means that P(*n*+1) is true, and so P(*t*) is true. This is a contradiction, since *t* was in S. Therefore, S is empty.

> I like (have started liking) to jot down whatever I feel like.. (smile)

reference: http://en.wikipedia.org/

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