Infima of two Real Sets

Theorem
Let $S$ and $T$ be real sets.

Let $S$ and $T$ admit infima.

Then:


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

Proof
Let:


 * $-S = \left\{ {-s: s \in S} \right\}$


 * $-T = \left\{ {-t: t \in T} \right\}$

Observe that:


 * $s \in S \iff -s \in -S$


 * $t \in T \iff -t \in -T$

We find:

Also see

 * Suprema of two Real Sets