Natural Numbers

Axiom: The empty set exists, which is to say that there is a set $\emptyset$ containing no elements, $\forall x, (x \notin \emptyset)$

Subset of Naturals with a Max Element has Bounded Difference

Suppose

S \subseteq N_{1}

has a max element

n

and min element

n

, and

D := {x - y : x \neq y \in S}

max (D) \leq m - n

Suppose

x - y \in D

since

x, y \in S

then

x \leq m

and

y \geq n

therefore:

x - y \leq m - y \leq m - n

thus

max (D) \leq m - n

as needed.