So i have $(X,d)$, a metric space. And function $F:X \times X \to \mathbb{R}$ given as $F(x,y)=d(x,y)$. So i want to prove that $F$ is Lipschitz on $X \times X$ if that space gets a metric $D_{1}$ defined as: $D_{1}=d(x_1,x_2)+d(y_1,y_2)$ So i know that i have to find $c>0$ such that it will work like this: $$\lvert F(x_1,x_2)-F(y_1,y_2)\rvert\le c \cdot D_1(x,y)$$ for any $(x,y)\in X\times X$
Any help would be appreciated.