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

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.

• You can not separate the $||x+y||$ like you did. Triangle inequality gives $||x+y||\leq||x||+||y||$, not $\geq$. To solve the exercise use the reverse triangle inequality: $||x||\leq||x-y||+||y||$, so $||x-y||\geq||x||-||y||$. Oct 17, 2018 at 23:27
• In the textbook I am using this is called the triangle inequality so I'm not sure I am aloud to do that. Oct 17, 2018 at 23:30
• You can prove the reverse triangle inequality yourself. It is only a few deductions. Oct 17, 2018 at 23:32

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