Basic Analysis I — Real Analysis · Problem 1.2.9

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · supremum-infimum

Problem

Let

A

and

B

be two nonempty bounded sets of real numbers. Let

C := \{a + b : a \in A, b \in B\}

. Show that

C

is a bounded set and that

\sup C = \sup A + \sup B \quad \text{and} \quad \inf C = \inf A + \inf B.