Prove that every Non-Empty set of Positive Integers contains a "Least Element".

Let S be the set. Since S is non empty, it has at least one element, which we call x. Since S has positive integers only, then x is integer and x is larger than 0. Let T = S intersected with {1, 2, .., x}. T is finite (it has at most x elements) and is non-empty (it contains at least the element x); thus T has a least element. The least element of T is also the least element of S. qed.

Posted by val on 2003-05-13 05:15:48