Are $f,g$ Lipschitz if there is $K\geq 0$ s.t. $d(f(x) , f(y)) + d(g(x), g(y)) \leq K d'(x,y)$ for all $x,y$?

Question

Assume that $f,g : X \to Y$ are functions between metric spaces. Assume that there is $K \geq 0$ such that $d_Y(f(x) , f(y)) + d_Y(g(x), g(y)) \leq K d_X(x,y)$ for all $x,y \in X$. Does this mean that $f,g$ must be Lipschitz? If so, what is the easiest way to see it?

EDIT: I changed a sign, this alters the question completely. But it now reads correctly.

Answer 1

No. Take $f$ any non-Lipschitz function, and take $g = f$.

Edit: The answer to the corrected question, as already given in comments by @orangeskid, is yes: if $d(f(x), f(y)) + d(g(x), g(y)) \leq K d(x,y)$, then a fortiori since metrics are non-negative, $d(f(x), f(y))\leq K d(x,y)$, so $f$ is Lipschitz. The same holds of course for $g$.