06 - Min-Max Property of Subsets of ℤ

Min-Max Property of subsets of $\mathbb{Z}$

Let $S$ be a non-empty subset of $\mathbb{Z}$

1. If $S$ is bounded below, then $S$ has a minimum

2. If $S$ is bounded above, then $S$ has a maximum

Proof - $(1)$

Assume that $S$ is bounded below
By completeness (applied to $\{-s:s\in S\}$), $S$ has an infimum

By approximation property (with $ϵ=1$), there is

$$n\in S\text{ such that }\inf S\le n<\inf S+1$$

If $n=\inf S$ then $\inf S\in S$ so there is a minimum

Otherwise, suppose for a contradiction, that $\inf S<n$
Suppose $n=\inf S+\delta$, where $0<\delta <1$

By the approximation property (with $ϵ=\delta$) there is

$$m\in S\text{ such that }\inf S\le m<\inf S+ϵ=n$$

Now $m<n$ so $n-m>0$ but as $n-m$ is an integer then $n-m\ge 1$
So

$$n\ge m+1\ge \inf S+1$$

which is a contradiction hence

$$n=\inf S\in S\text{ so }\inf S=\min S$$

Proof - $(2)$ - similar to $(1)$