Absolute Value

Overview

Let $x\in \mathbb{R}$. The absolute value of $x$, denoted $|x|$, is defined as $|x|=\{\begin{array}{ll}x& \text{if }x\ge 0\\ -x& \text{if }x\le 0\end{array}$

Triangle Inequality

Let $x,y\in \mathbb{R}$. Then the triangle inequality of $\mathbb{R}$ states $|x+y|\le |x|+|y|$

Bibliography

• Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
• “Triangle Inequality.” In Wikipedia, July 1, 2024. https://en.wikipedia.org/w/index.php?title=Triangle_inequality.