Condition for Supremum of Subset to equal Supremum of Set

Lemma
Let $S$ be a real set.

Let $T$ be a subset of $S$.

Let $S$ and $T$ admit suprema.

Then:


 * $\sup T = \sup S \iff \forall \epsilon \in \R_{>0}: \forall s \in S: \exists t \in T: s < t + \epsilon$

Necessary Condition
Let $\sup T = \sup S$.

The aim is to establish that $\forall \epsilon \in \R_{>0}: \forall s \in S: \exists t \in T: s < t + \epsilon$.

We have $\sup S \le \sup T$ as $\sup S = \sup T$.

The result follows by Suprema of two Real Sets.

Sufficient Condition
Let $\forall \epsilon \in \R_{>0}: \forall s \in S: \exists t \in T: s < t + \epsilon$.

The aim is to establish that $\sup T = \sup S$.

Observe that $T$ is non-empty as:


 * an empty set does not have a uniquely defined supremum


 * $T$ has a (uniquely defined) supremum

Accordingly, $T$ is a non-empty subset of $S$.

We have: