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

For any set with one element, that element is the least element

For any set with two elements, the smaller element is the least element.

Consider set S {a, b, n1, n2, ... nx} with x+2 (x >= 1)elements. Let c be the smaller of a and b. A new and smaller set T with x+1 elements can be defined as {c, n1, n2, ...nx}. Since c is the smaller of a and b, the least element was not removed which implies the sets S and T have the same least element.

For any set R of k elements with k > 2, this argument can be applied k-2 times to create a set of two elements with the same least element as R.