Exercise 866Calculus › Multivariable calculus › ContinuityL1Added 3 Nov 2026

Norm and distance are continuous

#norm#distance#continuity

Statement

Show that the Euclidean norm $\|\cdot\|:\mathbb{R}^n\to\mathbb{R}$ is continuous.

Hint

Use the reverse triangle inequality: $|\|x\|-\|y\||\le\|x-y\|$ .

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Inequality. By the reverse triangle inequality, for any $x,y$ : $|\|x\|-\|y\||\le\|x-y\|$ .

Conclusion. $\|x_n\|\to\|x\|$ whenever $x_n\to x$ : the norm is continuous.