1. ## 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!

2. ## 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.

3. ## Re: Smallest Element proof

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?

4. ## Re: Smallest Element proof

this is supposed to use the Well-Ordering Principle as well as "construction," which I am not sure about

5. ## 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.

6. ## Re: Smallest Element proof

Why can't 'b' be negative?

7. ## Re: Smallest Element proof

Originally Posted by jstarks44444
Why can't 'b' be negative?
It can; this does not change anything.