Distance between Element and Subset is Nonnegative

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.