# Smallest Element proof

• Jan 11th 2012, 07:24 PM
jstarks44444
Smallest Element proof
Hey all, need some help with the following proof:

Let A be a non-empty subset of Z and b ∈ Z, such that for each a ∈ A, b <= a. Then A has a smallest element.

All help appreciated!
• Jan 14th 2012, 08:19 AM
HallsofIvy
Re: Smallest Element proof
Using what theorems? You can use either induction or the "well ordered" property of the positive integers to prove this. They are equivalent but I don't know which you have to use. In either case, I think I would start by looking at B= {a- b| a∈ A}, a set of positive integers.
• Jan 14th 2012, 10:09 AM
emakarov
Re: Smallest Element proof
Quote:

Originally Posted by jstarks44444
Let A be a non-empty subset of Z and b ∈ Z, such that for each a ∈ A, b <= a. Then A has a smallest element.

Isn't b the smallest element by definition?
• Jan 15th 2012, 08:04 AM
jstarks44444
Re: Smallest Element proof
this is supposed to use the Well-Ordering Principle as well as "construction," which I am not sure about
• Jan 15th 2012, 10:30 AM
emakarov
Re: Smallest Element proof
Ah, yes, b does not have to be an element of A. Then, as post #2 suggested, you can consider the set B = {a - b | a ∈ A}, which is a subset of natural numbers and thus has the smallest element.
• Jan 15th 2012, 08:18 PM
jstarks44444
Re: Smallest Element proof
Why can't 'b' be negative?
• Jan 16th 2012, 02:06 AM
emakarov
Re: Smallest Element proof
Quote:

Originally Posted by jstarks44444
Why can't 'b' be negative?

It can; this does not change anything.