Distance from Subset to Supremum

Theorem
Let $S \subseteq \R$ be a subset of the real numbers.

Suppose that the supremum $\sup S$ of $S$ exists.

Then:


 * $\map d {\sup S, S} = 0$

where $\map d {\sup S, S}$ is the distance between $\sup S$ and $S$.

Proof
By Distance between Element and Subset is Nonnegative:


 * $\map d {\sup S, S} \ge 0$

By definition of supremum:


 * $\forall \epsilon > 0: \exists s \in S: \map d {\sup S, s} < \epsilon$

meaning that, by nature of the infimum and the definition of $\map d {\sup S, S}$:


 * $\forall \epsilon > 0: \map d {\sup S, S} < \epsilon$

Together, these two observations lead to the conclusion that:


 * $\map d {\sup S, S} = 0$

as desired.