Let $h: (\mathbb{R},d_2) \to (\mathbb{R},d_1)$ be a function where $d_1$ is the usual metric and $d_2: \mathbb{R} \times \mathbb{R} $ is a metric given by $d_2(x,y):=\min\{1, |x-y|\}$.
Show that there doesn't exist a bijective function $h$ so that $d_1(h(x),h(y))=d_2(x,y)$.
I've tried to think about this question in various ways but nothings seems to get me going. I figured I would only have to show that if such a function exists it can't be injective or surjective. I also thought of showing that the domain and range aren't of equal size. But I wouldn't even know where to start with that. Could somebody help me out?