Isometries between metric spaces

Question

Could you help me with the next exercise please:

Let $(X,d_{x})$ and $(Y,d_{y})$ be metric spaces. If there is a function $f:X \rightarrow Y$ such that $d_{x}(x,x')=d_{y}(f(x),f(x'))$ for each $x, x' \in X$, prove that then $X$ is isometric to a subspace of $Y$.

I have tried applying the definition, it really seems to me that this is the definition of isometry, but they tell me that you have to do something more to be able to finish the exercise. I would very much appreciate your help please, thanks.

Answer 1

• Show that $f$ is injective
• Therefore $f$ is a bijection between $X$ and its image $f(X)$ (which is a subset of $Y$).
• Verify this bijection is an isometry.