# Absolute Value is Bounded Below by Zero

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## 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.

$\blacksquare$

## 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.

## Also see

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.14$: Modulus