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:

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 the empty set does not admit a supremum (in $\R$).

We have: