Triangle inequality

Triangle inequality (metric form): In a metric space $(X,d)$, for all $x,y,z\in X$, $d(x,z)\le d(x,y)+d(y,z).$

Triangle inequality (norm form): In a normed vector space $(V,\|\cdot\|)$, for all $u,v\in V$, $\|u+v\|\le \|u\|+\|v\|.$

The triangle inequality is the foundational estimate behind most $\varepsilon$–arguments in analysis, including limit uniqueness, Cauchy criteria , and continuity estimates.

Examples:

• In $\mathbb{R}$ with $d(x,y)=|x-y|$, the metric triangle inequality is $|x-z|\le |x-y|+|y-z|$.
• In $\mathbb{R}^n$ with the Euclidean norm , $\|u+v\|\le \|u\|+\|v\|$ follows from Cauchy–Schwarz .