Proving a function is Lipschitz for a metric, and investigating Lipschitz continuity for a different metric

Question

So i have a metric space $(X,d)$ and the function $F: X \times X \rightarrow \mathbb{R}$ defined as: $F(x,y)= d(x,y)$.

Now i have to prove, that F is Lipschitz continuous on $X \times X$ if the metric in $X \times X$ is $D_1((x_1,y_1),(x_2,y_2))=d(x_1,x_2)+d(y_1,y_2)$

So i know i need to somehow find a $c\in\mathbb{R}$ such that this will be true:

$$ d_2(f(x_1,y_1),f(x_2,y_2)) \leq c \cdot D_1((x_1,y_1)(x_2,y_2)) $$ $$d_2(f(x_1,y_1),f(x_2,y_2))= d_2(d(x_1,y_1),d(x_2,y_2))$$

So should ijust use the definition of $d_2$ and then i will find that $c$?

Also i have the second thing to do, i need to investigate same function $F$, but the metric D_1 is swapped with this metric:

$D_{\infty}((x_1,y_1)(x_2,y_2))=max${$d(x_1,x_2),d(y_1,y_2)$}

My first idea is that it won't be, but i don't know how to prove it that way.

Any help regarding both problems would be appreciated.

Answer 1

$F(x_1,y_1)-F(x_2,y_2)=d(x_1,y_1)-d(x_2,y_2)≤d(x_1,x_2)+d(x_2,y_1)-d(x_2,y_2)≤d(x_1,x_2)+d(y_1,y_2)≤2max[{d(x_1,x_2),d(y_1,y_2)}]$. The other side is dealt with similarly.These inequalities solve the problem I hope.