# Prove that Prove $\left\lvert \left\lVert x \right\rVert - \left\lVert y \right\rVert \right\rvert \le \left\lVert x-y \right\rVert$

This is Exercise 3.1.4 from Economic Dynamics, Theory and Computation by John Stachursky. Key definitions for the exercise are his definition of norm and metric I believe.

Let Prove $\left\lVert \cdot \right\rVert$ be a norm on $\Bbb R^k$. Show that for any $x,y \in \Bbb R ^k$ we have $\left\lvert \left\lVert x \right\rVert - \left\lVert y \right\rVert \right\rvert \le \left\lVert x-y \right\rVert$

He does define beforehand the absolute value, and the righthand side of the equation is the definition of metric, acording to the book, but I don't know how to proceed.

$$\|x\|=\|x-y+y\| \leq \|x-y\|+\|y\|$$ $$\implies \|x\| -\|y\| \leq \|x-y\|.$$ Swapping $x$ and $y$ gives the modulus.
• @MartinValdez Write down what I wrote, and then below it put $y$ where $x$ was and vice-versa. Everything from the beginning still holds and you'll notice that the left-hand side of the last equation is going to be pleasant. – Aloizio Macedo Dec 13 '15 at 21:51