Does such a metric exist? Does a metric $d: (-1,1) \times (-1,1) \to \mathbb{R}_+$ exist such that the open interval $(-1,1)$ is complete (with respect to $d$) and that $d$ is equivalent to the usual Euclidean metric, $d_u$?

I'm having trouble figuring this out. First I thought that such a metric couldn't exist. I figured maybe we could define some function $f:((-1,1),d) \to (\mathbb{R}, d_u)$ and then try a cauchy sequence or something, but I really have no idea.. to add to all of this there are people saying that such a metric does indeed exist

There is indeed such a metric, and you're on the right track thinking about ways to translate from $(-1, 1)$ to $\mathbb{R}$. HINT: think about the function $\arctan x$ . . .
