Basic Analysis I — Real Analysis · Problem 1.3.8

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · bounded-functions

Problem

Suppose

D

is nonempty,

f : D \to \mathbb{R}

and

g : D \to \mathbb{R}

are bounded functions, and

\alpha \in \mathbb{R}

. a) Show that

\alpha f : D \to \mathbb{R}

defined by

(\alpha f)(x) := \alpha f(x)

is a bounded function. b) Show that

f + g : D \to \mathbb{R}

defined by

(f + g)(x) := f(x) + g(x)

is a bounded function.