Hey all, I would appreciate some help with the following proof:
Let A be a nonempty subset of Z and b is an element of the Integers, such that for each a which is an element of A, b <= a. Then A has a smallest element.
I think the Well-Ordering Principle is supposed to be used here..thanks for the help!
February 17th 2011, 09:35 PM
If b is in A, then b is the smallest element. Otherwise, b+1 is a lower bound on A, and you repeat the argument--if b+1 is in A, then b+1 is the smallest element, otherwise b+2 is a lower bound on A. This won't continue forever, because A is nonempty. Pick any a in A, and this process will terminate in no more than b-a steps.