Basic Analysis I — Real Analysis · Problem 1.1.14

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · ordered-sets

Problem

Let

S := \{a_k : k \in \mathbb{N}\} \cup \{b_k : k \in \mathbb{N}\}

, ordered such that

a_k < b_j

for every

k

and

j

a_k < a_m

whenever

k < m

, and

b_k > b_m

whenever

k < m

. a) Show that

S

is an ordered set. b) Show that every subset of

S

is bounded (both above and below). c) Find a bounded subset of

S

that has no least upper bound.