Distance between Element and Subset is Nonnegative

Theorem
Let $\left({M, d}\right)$ be a metric space.

Let $x \in M$ and $S \subseteq M$.

Then:


 * $d \left({x, S}\right) \ge 0$

where $d \left({x, S}\right)$ is the distance between $x$ and $S$.

Proof
By definition of the distance between $x$ and $S$:


 * $d \left({x, S}\right) = \displaystyle \inf_{s \in S} d \left({x, s}\right)$

From the metric space axioms:


 * $\forall s \in M: d \left({x, s}\right) \ge 0$

Hence by the nature of the infimum:


 * $d \left({x, S}\right) \ge 0$

as desired.