Combination Theorem for Bounded Continuous Real-Valued Functions/Difference Rule

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Theorem

Let $\struct{S, \tau}$ be a topological space.

Let $\R$ denote the real number line.

Let $f, g :S \to \R$ be bounded contiuous real-valued functions.

Let $f - g : S \to \R$ be the pointwise difference of $f$ and $g$, that is, $f - g$ is the mappping defined by:

$\forall s \in S : \map {\paren{f - g} } s = \map f s - \map g s$

Then:

$f - g$ is a bounded coninuous real-valued function

Proof

Follows from:

• Difference Rule for Bounded Real-Valued Functions
• Difference Rule for Continuous Real-Valued Functions

$\blacksquare$