Show that | $\left\lVert x \right\rVert - \left\lVert y \right\rVert$ | $\leq$ $\left\lVert x - y \right\rVert$ in a normed space.

Question

Show that | $\left\lVert x \right\rVert - \left\lVert y \right\rVert$ | $\leq$ $\left\lVert x - y \right\rVert$ for any x and y in a normed space.

Here is my attempt:

Take the triangle inequality \begin{align*} \left\lVert x + y \right\rVert &\leq \left\lVert x + y \right\rVert + \left\lVert y \right\rVert \\ &= \left\lVert x - y + y \right\rVert + \left\lVert y \right\rVert \\ &\leq \left\lVert x - y\right\rVert + \left\lVert y \right\rVert + \left\lVert y \right\rVert. \end{align*}

Then we move the two $\left\lVert y \right\rVert$'s to the other side \begin{align*} \left\lVert x-y \right\rVert &\geq \left\lVert x + y \right\rVert - \left\lVert y \right\rVert - \left\lVert y \right\rVert \\ &\geq \left\lVert x \right\rVert + \left\lVert y \right\rVert - \left\lVert y \right\rVert - \left\lVert y \right\rVert\\ &\geq \left\lVert x \right\rVert - \left\lVert y \right\rVert.\end{align*}

I am not sure if I can separate the $\left\lVert x + y \right\rVert$ into $\left\lVert x \right\rVert + \left\lVert y \right\rVert$ like I did and I am not sure how to get absolute values so any hints would be great.

Answer 1

Assume that $\| x\|\geq \|y\|$. Then $$|\|x\|-\|y\|| =\|x\|-\|y\|\leq \|x-y\|$$