Absolute Value is Bounded Below by Zero

Theorem
Let $x \in \R$ be a real number.

Then the absolute value $\size x$ of $x$ is bounded below by $0$.

Proof
Let $x \ge 0$.

Then $\size x = x \ge 0$.

Let $x < 0$.

Then $\size x = -x > 0$.

The result follows.

Also see

 * Absolute Value on Ordered Integral Domain is Strictly Positive except when Zero