Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces, $f:(X,d_X) \rightarrow (Y,d_Y)$ be an isometry. Then $f$ is Lipchitz-continuous.
Suppose that $f$ is an isometry. Then for all $x_1,x_2$ in $X$, we have $d_Y(f(x_1),f(x_2)) = d(x_1,x_2)$...
From here, how do I prove that there exists $k>0$ such that for all $x_1,x_2 \in X,$ $d_Y(f(x_1),f(x_2))\leq kd_X(x_1,x_2)$