Combination Theorem for Bounded Real-Valued Functions/Negation Rule

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Theorem

Let $S$ be a set.

Let $\R$ denote the real number line.

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

Let $-f : S \to \R$ denote the pointwise negation of $f$, that is, $-f$ denotes the mapping defined by:

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

Then:

$-f$ is a bounded real-valued function

Proof

By definition of bounded real-valued function

$\exists M \in \R_{\ge 0} : \forall s \in S : \size{\map f s} \le M$

We have:

\(\ds \forall s \in S: \, \)

\(\ds \size{\map {\paren{-f} } s}\)

\(=\)

\(\ds \size{-\map f s}\)

Definition of Absolute Value of Real-Valued Function

\(\ds \)

\(=\)

\(\ds \size{\map f s}\)

Absolute Value of Negative

\(\ds \)

\(\le\)

\(\ds M\)

Definition of Bounded Real-Valued Function

It follows that $-f$ is a bounded real-valued function by definition.

$\blacksquare$