Norm is Continuous

Theorem

Let $\left({V, \left\Vert{\cdot}\right\Vert}\right)$ be a normed vector space.

Then the mapping $x \mapsto \left\Vert{x}\right\Vert$ is continuous.

Here, the metric used is the metric $d$ induced by $\left\Vert{\cdot}\right\Vert$.

Proof

Since $\left\Vert{x}\right\Vert = d \left({x, \mathbf 0}\right)$, the result follows directly from Metric is Continuous.

$\blacksquare$