Combination Theorem for Bounded Continuous Real-Valued Functions/Sum 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 addition 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:

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

$\blacksquare$