Basic Analysis I — Real Analysis · Problem 1.1.13

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · ordered-sets

Problem

Let

\mathbb{N}_\infty := \mathbb{N} \cup \{\infty\}

, where the elements of

\mathbb{N}

are ordered in the usual way amongst themselves, and

k < \infty

for every

k \in \mathbb{N}

. Show

\mathbb{N}_\infty

is an ordered set and that every subset

E \subset \mathbb{N}_\infty

has a supremum in

\mathbb{N}_\infty

(make sure to also handle the case of an empty set).