Basic Analysis I — Real Analysis · Problem 1.2.15

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · dedekind-cuts

Problem

a) Prove that given

y \in \mathbb{R}

, we have

\sup\{x \in \mathbb{Q} : x < y\} = y

. b) Let

A \subset \mathbb{Q}

be a set that is bounded above such that whenever

x \in A

and

t \in \mathbb{Q}

with

t < x

, then

t \in A

. Further suppose

\sup A \notin A

. Show that there exists a

y \in \mathbb{R}

such that

A = \{x \in \mathbb{Q} : x < y\}

. A set such as

A

is called a Dedekind cut. c) Show that there is a bijection between

\mathbb{R}

and Dedekind cuts. Note: Dedekind used sets as in part b) in his construction of the real numbers.