Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above/Corollary

Corollary to Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above
Let $S$ be a subset of the real numbers $\R$.

Let $T = \set {x \in \R: -x \in S}$ be the set of negatives of the elements of $S$.

Then:
 * $S$ is bounded above


 * $T$ is bounded below.
 * $T$ is bounded below.

Proof
Let $V$ be the set defined as:


 * $V = \set {x \in \R: -x \in T}$

From Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above:


 * $T$ is bounded below


 * $V$ is bounded above.
 * $V$ is bounded above.

Then we have:

The result follows.