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November 4th, 2014, 10:06 AM   #1
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Proof difference of two natural numbers

Hey, I got this question in my Calculus class, and I'm completely stuck: Assume N (natural numbers) is the intersection of all the inductive sets of R (real numbers). prove that ∀x,y ∈ N, if x>y, then x-y ∈ N. I do not ask for a solution - only for a hint on how to proceed - I've checked the web and calculus books for any help, and found none. Like other proofs by my instructor, we'll construct a group B = {x∈ N, ∀y∈N, such that if y<x then x-y ∈N} And then prove B is inductive set, and since N is the intersection of all the inductive sets then N=B, and then we've proven that N has the properties of group B. I can't seem to prove that B is an inductive set (specifically, proving that if x∈B, then x+1∈B)

p.s.
Here's the definition of inductive set A:
(1 ∈ A) ∧ (∀x)(X ∈ A → x+1 ∈ A).

Last edited by godingly; November 4th, 2014 at 10:11 AM.
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