Combination Theorem for Bounded Continuous Real-Valued Functions/Minimum Rule

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Theorem

Let $\struct{S, \tau}$ be a topological space.

Let $\R$ denote the real number line.

Let $f, g :S \to \R$ be bounded continuous real-valued functions.

Let $f \wedge g$ denote the pointwise minimum of $f$ and $g$, that is, $f \wedge g$ is the mapping defined by:

$\forall s \in S : \map {\paren{f \wedge g} } s = \min \set{\map f s, \map g s}$

Then:

$f \wedge g$ is a bounded continuous real-valued function

Proof

Follows from:

• Minimum Rule for Bounded Real-Valued Functions
• Minimum Rule for Continuous Real-Valued Functions

$\blacksquare$