proof by contradictin / using the well ordering property

[rayancaleb]

Hi,



I am having trouble understanding this proof that every integer from 2 onwards can expressed as a product of primes:

Assume the negation of the statement is true (proof by contradiction):Negation of the statement: Every integer from 2 onwards cannot expressed as a product of primes.

Let S be the set of integers from 2 onwards which cannot be expressed as a product of primes.By the well ordering property, this set has a least element s, say.The proof then goes on to show that s is a product of numbers less than s which are prime. So s is a product of primes and I accept this.



The proof then states "So we have a contradiction and s is not S therefore S is empty."

So the proof shows that s is not an element of S.

How does this prove that S is empty?



Please Help.



Thanks,



I didn't find the right solution from the internet.

References:



mathforum.org/kb/forumcategory.jspa?categoryID=15



Tutoring Solution Video