How do I show that in a normed space $| (\|x\|-\|y\|) | \leq \|x-y\|$

Question

How do I show that in a normed space $| (\|x\|-\|y\|) | \leq \|x-y\|$

I need to use normed space axioms but I'm unable to figure it out. I think the scalar axiom and triangle inequality are helpful.

Answer 1

By triangle inequality: $$\|x\| = \|x - y + y\| \leq \|x - y\| + \|y\|$$ implies $\|x\| - \|y\| \leq \|x - y\|$. And $$\|y\| = \|y - x + x\| \leq \|y - x\| + \|x\|$$ implies $\|y\| - \|x\| \leq \|y - x\| = \|x - y\|$.

Therefore $$\left|\|x\| - \|y\|\right| \leq \|x - y\|.$$

Answer 2

$\left \| x \right \|\leq \left \| x-y \right \|+\left \| y \right \|$

$\left \| y \right \|\leq \left \| x-y \right \|+\left \| x \right \|$

so

$\left \| x \right \|-\left \| y \right \|\leq \left \| x-y \right \|$

and

$\left \| y \right \|-\left \| x \right \|\leq \left \| x-y \right \|$

which is what you want.