Absolute Value Bounded Below by Zero

Theorem

Let $x$ be a real number, i.e. that $x \in \R$.

Then the absolute value $\left|{x}\right|$ of $x$ is bounded below by $0$.

Proof

• Let $x \ge 0$.

Then $\left|{x}\right| = x \ge 0$.

• Let $x < 0$.

Then $\left|{x}\right| = -x > 0$.

The result follows.

$\blacksquare$

Notes

This result applies also to the set of integers $\Z$ and rational numbers $\Q$.

In that context the result is usually included as part of the field of number theory as well as that of analysis.

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.14$