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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.

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$$\|x\|=\|x-y+y\| \leq \|x-y\|+\|y\|$$ $$ \implies \|x\| -\|y\| \leq \|x-y\|. $$ Swapping $x$ and $y$ gives the modulus.

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  • $\begingroup$ Good lord this is so obvios I can't believe I didn't see it What do you mean by swapping? $\endgroup$ – Martin Valdez Dec 13 '15 at 21:48
  • $\begingroup$ @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. $\endgroup$ – Aloizio Macedo Dec 13 '15 at 21:51
  • $\begingroup$ Ok I see it now, thanks! $\endgroup$ – Martin Valdez Dec 13 '15 at 21:53

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