Constant Real-Valued Function is Bounded

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Theorem

Let $S$ be a set.

Let $\R$ denote the real number line.

Let $c \in \R$.

Let $c_{\R^S} : S \to R$ be the constant mapping defined by:

$\forall s \in S : \map {c_{\R^S}} s = c$

Then $c_{\R^S}$ is a bounded real-valued function.

Proof

We have:

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

\(\ds \size{\map {c_{\R^S} } s}\)

\(=\)

\(\ds \size c\)

Definition of Constant Mapping

\(\ds \)

\(\le\)

\(\ds \size c\)

It follows that $c_{\R^S}$ is a bounded real-valued function by definition.

$\blacksquare$