Combination Theorem for Bounded Real-Valued Functions/Sum 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

By definition of bounded real-valued function

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

and

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

Let $M = M_f + M_g$.

We have:

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

\(\ds \size{\map {\paren{f + g} } s}\)

\(=\)

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

Definition of Pointwise Addition of Real-Valued Functions

\(\ds \)

\(\le\)

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

Triangle Inequality for Real Numbers

\(\ds \)

\(\le\)

\(\ds M_f + M_g\)

Definition of Bounded Real-Valued Function

\(\ds \)

\(=\)

\(\ds M\)

Definition of $M$

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

$\blacksquare$