Prove that every Non-Empty set of Positive Integers contains a "Least Element".
Let S denote a nonempty set of positive integers with no least element.
1 is not an element of this set, since there is no positive integer less than 1, and therefore it would be a least value.
Assume that all integers in the range [1, n] are not part of S. Then n+1 is not a member of S, because it would be a least element of S.
By induction, every positive integer is not a member of S. Since S consists only of positive integers, it is the empty set. However, it was given that S is not empty. Thus, by contradiction, every nonempty set of positive integers must have a least element.
Proof by Contradiction
