Lipschitz Continuity Defintion: Given two metric spaces $(X, d_X)$ and $(Y, d_Y)$, where $d_X$ denotes the metric on the set $X$ and $d_Y$ is the metric on set $Y$, a function $f:X \to Y$ is called Lipschitz continuous if there exists a real constant $K \geq 0$ such that, for all $x_1$ and $x_2$ in $X$,
$d_Y (f(x_1),f(x_2))\leq Kd_X(x_1,x_2)$
Let's say that I want to bound $\Vert f(x_1,x_2)-f(\hat{x}_1,\hat{x}_2)\Vert _2$ with the Lipschitz ineqaulity, where the function $f:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n$ is Lipschitz, and the Euclidian norm is denoted by $\Vert\cdot\Vert _2$. So following the above inequality I get $$\Vert f(x_1,x_2)-f(\hat{x}_1,\hat{x}_2)\Vert _2\leq K\Vert (x_1,x_2)-(\hat{x}_1,\hat{x}_2)\Vert _m,$$ where $\Vert\cdot\Vert _m$ denotes the metric for $\mathbb{R}^n\times\mathbb{R}^2$, but I am unsure of what metric to use? Do I use the Frobenious norm, or something else? Is this open to choice? If so, how do I choose? What are the rules? I can really use some clarity on this. Thanks