Is an isometric and bijective mapping between two metric spaces complete?

Question

If I have the two metric spaces $(X,d_x)$ and $(Y,d_y)$ with the mapping $f : X \to Y$ that is both an isometry and bijection between X and Y.

How do I show that $(Y,d_y)$ is complete iff $(X,d_x)$ is complete?

Answer 1

This is a matter of understanding the concept of isometry as just being a compatible relabeling of the elements. Assume $Y$ is complete. Take a Cauchy sequence in $X$. The isometry gives you a sequence in $Y$. Show it is Cauchy, and hence converges. This gives you a limit in $Y$. You can apply the inverse of the isometry to get a point in $X$, and show that the original sequence converges to this point.

Answer 2

Why would they be complete?

Consider e.q. $\Bbb Q\times\{0\}$ and $\Bbb Q\times\{1\}$ as subspaces of the real plane and $f:=(x,0)\mapsto (x,1)$.