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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.

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  • $\begingroup$ 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 $||x-y||\geq||x||-||y||$. $\endgroup$ – SmileyCraft Oct 17 '18 at 23:27
  • $\begingroup$ In the textbook I am using this is called the triangle inequality so I'm not sure I am aloud to do that. $\endgroup$ – Polectio Oct 17 '18 at 23:30
  • $\begingroup$ You can prove the reverse triangle inequality yourself. It is only a few deductions. $\endgroup$ – SmileyCraft Oct 17 '18 at 23:32
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Assume that $\| x\|\geq \|y\|$. Then $$|\|x\|-\|y\|| =\|x\|-\|y\|\leq \|x-y\|$$

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