Distance between Element and Subset is Nonnegative
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Theorem
Let $\struct {M, d}$ be a metric space.
Let $x \in M$ and $S \subseteq M$.
Then:
- $\map d {x, S} \ge 0$
where $\map d {x, S}$ is the distance between $x$ and $S$.
Proof
By definition of the distance between $x$ and $S$:
- $\map d {x, S} = \ds \inf_{s \mathop \in S} \map d {x, s}$
From the metric space axioms:
- $\forall s \in M: \map d {x, s} \ge 0$
Hence by the nature of the infimum:
- $\map d {x, S} \ge 0$
as desired.
$\blacksquare$