Constant Real-Valued Function is Bounded
Jump to navigation
Jump to search
This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code. |
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.
$\blacksquare$