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

if and only if:

$T$ 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

if and only if:

$V$ is bounded above.

Then we have:

\(\ds V\)

\(=\)

\(\ds \set {x \in \R: -x \in T}\)

\(\ds \)

\(=\)

\(\ds \set {x \in \R: -\paren {-x} \in S}\)

\(\ds \)

\(=\)

\(\ds \set {x \in \R: x \in S}\)

\(\ds \)

\(=\)

\(\ds S\)

The result follows.

$\blacksquare$