Basic Analysis I — Real Analysis · Problem 1.3.9

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · bounded-functions

Problem

Let

f : D \to \mathbb{R}

and

g : D \to \mathbb{R}

be functions with

D

nonempty,

\alpha \in \mathbb{R}

, and recall what

f + g

and

\alpha f

means from the previous exercise. a) Prove that if

f + g

and

g

are bounded, then

f

is bounded. b) Find an example where

f

and

g

are both unbounded, but

f + g

is bounded. c) Prove that if

f

is bounded but

g

is unbounded, then

f + g

is unbounded. d) Find an example where

f

is unbounded but

\alpha f

is bounded.