Distance from Subset to Supremum
Jump to navigation
Jump to search
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.
$\blacksquare$