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:


 * $d \left({\sup S, S}\right) = 0$

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

Proof
By Distance between Element and Subset is Nonnegative:


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

By definition of supremum:


 * $\forall \epsilon > 0: \exists s \in S: d \left({\sup S, s}\right) < \epsilon$

meaning that, by nature of the infimum and the definition of $d \left({\sup S, S}\right)$:


 * $\forall \epsilon > 0: d \left({\sup S, S}\right) < \epsilon$

Together, these two observations lead to the conclusion that:


 * $d \left({\sup S, S}\right) = 0$

as desired.