Basic Analysis I — Real Analysis · Problem 1.1.9

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · ordered-sets

Problem

Let

S

be an ordered set and

A

is a nonempty subset such that

\sup A

exists. Suppose there is a

B \subset A

such that whenever

x \in A

there is a

y \in B

such that

x \leq y

. Show that

\sup B

exists and

\sup B = \sup A