If $x\mapsto f(x)=x\Vert x\Vert$, then $f$ is locally Lipschitz

Question

Suppose \begin{align}f:\Bbb{R}^{n}\to\Bbb{R}^n\end{align} \begin{align}x\mapsto f(x)=x\Vert x\Vert\end{align} I want to prove that $f$ is locally Lipschitz.

MY WORK

Let $x,y\in\Bbb{R}^{n}, $then \begin{align}\Vert f(x)-f(y)\Vert=\Vert x\Vert x\Vert -y\Vert y\Vert\Vert\end{align} I got stuck at this point, any help please?

Answer 1

Based on John Ma's hint,

Let $x,y\in\Bbb{R}^{n}, $then \begin{align}\Vert f(x)-f(y)\Vert&=\Vert x\Vert x\Vert -y\Vert y\Vert\Vert\\ &=\Vert x\Vert x\Vert -x\Vert y\Vert +x\Vert y\Vert -y\Vert y\Vert\Vert\\ &\leq \Vert x\Vert\Big\Vert\Vert x\Vert -\Vert y\Vert\Big\Vert +\Vert y\Vert\Vert x -y\Vert\\ &\leq \Vert x\Vert\Vert x -y\Vert +\Vert y\Vert\Vert x -y\Vert\\ &\leq \left(\Vert x\Vert+\Vert y\Vert\right)\Vert x -y\Vert\\ &= k\Vert x -y\Vert \end{align} and we are done!