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.