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!

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- Jan 11th 2012, 07:24 PMjstarks44444Smallest 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 AMHallsofIvyRe: 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 AMemakarovRe: Smallest Element proof
- Jan 15th 2012, 08:04 AMjstarks44444Re: 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 AMemakarovRe: 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 PMjstarks44444Re: Smallest Element proof
Why can't 'b' be negative?

- Jan 16th 2012, 02:06 AMemakarovRe: Smallest Element proof