Prove that $\Vert\cdot \Vert: X\to \Bbb{R},$ defined by $x\mapsto \Vert x \Vert,$ is continuous

Question

I want to prove that $\Vert\cdot \Vert: X\to \Bbb{R},$ defined by $x\mapsto \Vert x \Vert,$ is continuous, where $X$ is a normed linear space.

Here's what I've tried.

TRIAL

Let $\epsilon>0$, we seek $\delta$ such that $\Vert x_n-x \Vert<\delta,\;\forall\;n\geq N,$ for some $N$ implies $\Big|\Vert x_n\Vert-\Vert x \Vert \Big|<\epsilon,\;\forall\;n\geq N.$ Then,

\begin{align}\Big|\Vert x_n\Vert-\Vert x \Vert \Big|\leq \Vert x_n-x \Vert <\delta \end{align}
So, given any $\epsilon>0,$ take $\delta=\epsilon.$ Then, $\forall\;n\geq N,\;\Big|\Vert x_n\Vert-\Vert x \Vert \Big|<\epsilon,\;\forall\;n\geq N.$ Hence, we are done!

Please, I'm I right? If not, I need someone to help fine-tune the proof! Thanks

Answer 1

Given a normed, linear space $X$, the norm $\| \cdot \|$ satisfies the triangle inequality

$$\|x + y\| \leqslant \|x \| + \|y\|$$

Hence,

$$\|x\| = \|y + (x-y) \| \leqslant \|y\| + \|x - y\|, \\ \|y\| = \|x - (x-y) \| \leqslant \|x\| + \|-1(x - y)\| = \|x\| + \|(x - y)\|, $$

The first inequality implies $\|x\| - \|y \| \leqslant \|x-y\|$ and the second implies $\|x\| - \|y \| \geqslant -\|x-y\|$

Thus,

$$| \, \|x\| - \|y\| \, | \leqslant \|x - y \|$$

This proves (uniform) continuity since for all $x,y \in X$

$$\|x - y\| < \delta (= \epsilon) \implies | \, \|x\| - \|y\| \, | < \epsilon$$