Norm is Continuous

Theorem
Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space.

Then the mapping $x \mapsto \norm x$ is continuous.

Here, the metric used is the metric $d$ induced by $\norm {\,\cdot\,}$.

Proof
Since $\norm x = \map d {x, \mathbf 0}$, the result follows directly from Distance Function of Metric Space is Continuous.