Combination Theorem for Bounded Real-Valued Functions/Difference Rule

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Theorem

Let $S$ be a set.

Let $\R$ denote the real number line.

Let $f, g :S \to \R$ be bounded 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 real-valued function

Proof

From Pointwise Difference is Pointwise Addition with Negation:

$f - g = f + \paren{-g}$

where:

$-g$ denotes the pointwise negation of $g$

$f + \paren{-g}$ denotes the pointwise addition of $f$ and $-g$

From Negation Rule for Bounded Real-Valued Function:

$-g$ is a bounded real-valued function

From Sum Rule for Bounded Real-Valued Functions

$f + \paren{-g}$ is a bounded real-valued function

The result follows.

$\blacksquare$