Proof properties of vector norm

Question

How can I proof that for all vector norm on $ \mathbb{R} $ that $\left | \left \| x \right \|-\left \| y \right \| \right |\leq \left \| x-y \right \|$

Answer 1

Hint: For all real $x,y$, we have $x = (x-y)+y$. Apply the triangle inequality to this.

Answer 2

Notice that $||x|| - ||y|| \leq ||x-y|| $ as pointed out in the comments. Moreover,

$$ ||y|| \leq ||y-x||+||x|| \implies -||y-x|| \leq ||x||-||y||$$

Combining this with the first line, one observes that

$$ -||y-x|| \leq ||x||-||y|| \leq ||x-y|| $$

And this implies that

$$ | ||x|| - ||y|| | \leq ||x-y|| $$

the reversed triangle inequality

Answer 3

In case you didn't get the hints:

\begin{align*} \|x \| &= \|x-y +y \|\\ &\leq \|x-y \|+ \|y \| \end{align*}

and

\begin{align*} \|y \| &= \|y-x +x \|\\ &\leq \|y-x \|+ \|x \|\\ &= \|x-y \|+\| x\| \end{align*}

Hence, $\|x \|-\|y \| \leq \|x-y \|$ and $ \|x \|- \| y\| \geq -\|x -y \|$.

Thus, \begin{equation*} \big| \|x\| -\|y\| \big| \leq \|x-y \|.\end{equation*}