Basic Analysis I — Real Analysis · Problem 1.1.4

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · ordered-sets

Problem

Let

S

be an ordered set. Let

B \subset S

be bounded (above and below). Let

A \subset B

be a nonempty subset. Suppose all the infs and sups exist. Show that

\inf B \leq \inf A \leq \sup A \leq \sup B.