Basic Analysis I — Real Analysis · Problem 1.3.7

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · bounded-functions

Problem

Let

D

be a nonempty set. Suppose

f : D \to \mathbb{R}

and

g : D \to \mathbb{R}

are bounded functions. a) Show

\sup_{x \in D}\bigl(f(x) + g(x)\bigr) \leq \sup_{x \in D} f(x) + \sup_{x \in D} g(x) \quad \text{and} \quad \inf_{x \in D}\bigl(f(x) + g(x)\bigr) \geq \inf_{x \in D} f(x) + \inf_{x \in D} g(x).

b) Find an example (or examples) where we obtain strict inequalities.