Condition for Infimum of Subset to equal Infimum of Set

Lemma
Let $S$ be a real set.

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

Let $S$ and $T$ admit infima.

Then:


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

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

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

We have:

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

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

We have: