Absolute Value is Bounded Below by Zero
Jump to navigation
Jump to search
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