Reverse triangle inequality

Reverse triangle inequality: In a normed vector space $(V,\|\cdot\|)$, for all $u,v\in V$, $\bigl|\|u\|-\|v\|\bigr|\le \|u-v\|.$ Equivalently, $\|u\|\le \|v\|+\|u-v\|\quad\text{and}\quad \|v\|\le \|u\|+\|u-v\|.$

This inequality is frequently used to show continuity of the norm map and to transfer convergence of vectors to convergence of their norms.

Examples:

• In $\mathbb{R}$, the inequality becomes $\bigl||a|-|b|\bigr|\le |a-b|$.
• If $u_n\to u$ in a normed space, then $\|u_n\|\to \|u\|$ by the reverse triangle inequality .