Let $f(x,y) = (3x + 2y_1, y_1 - y_2)$, where $y = (y_1,y_2)$. Then $f$ satisfies a Lipschitz condition with $K=3$. For $$\vert f(x,y) - f(x,z)\vert=\vert2(y_1 - z_1), (y_1-z_1)-(y_2-z_2)\vert $$ $$\leq2\vert y_1-z_1\vert+\vert(y_1-z_1)-(y_2-z_2)\vert$$ $$\leq2\vert y_1-z_1\vert+\vert y_1-z_1\vert+\vert y_2-z_2\vert$$ $$\leq3\vert y-z\vert$$
Why is the last inequality true? If anything it should be the other way around, due to the triangle inequality.
They are a bit sloppy. It should be
\begin{align} |f(x, y) - f(x, z)| &= |2(y_1 - z_1), (y_1- z_1) - (y_2 - z_2)|\\ &= \sqrt{4(y_1-z_1)^2 + ((y_1-z_1) - (y_2- z_2))^2} \\ &\le \sqrt{4(y_1-z_1)^2 +2 (y_1-z_1)^2 + 2(y_2- z_2)^2} \\ &\le \sqrt 6 \sqrt{(y_1-z_1)^2 +(y_2- z_2)^2} = \sqrt 6 | (x, y) - (x, z)|. \end{align}
BTW, to check if $f$ is Lipschitz they should bound $|f(x, y) - f(w, z)|$ instead...