Zero is both Positive and Negative

Theorem
The number $0$ (zero) is the only (real) number which is both:
 * a positive (real) number

and
 * a negative (real) number.

Proof
Let $x$ be a real number which is both positive and negative.

Thus:
 * $x \in \set {x \in \R: x \ge 0}$

and:
 * $x \in \set {x \in \R: x \le 0}$

and so:
 * $0 \le x \le 0$

from which:
 * $x = 0$

Also defined as
In, $0$ is considered to be included in both the set of positive real numbers and the set of negative real numbers.

However, many sources consider $0$ to be neither positive nor negative.

Hence under that convention this result is no longer true.