Show that $X=(-1,1)$ with $d(x,y)=\left|\frac{x}{1-x^2} - \frac{y}{1-y^2}\right|$ is a complete metric space.

Question

On the space $X = (-1,1)$ with $d(x,y)=\left|\frac{x}{1-x^2} - \frac{y}{1-y^2}\right|$.

Prove that $(X,d)$ is complete metric space.

What I've tried: I considered a Cauchy sequence $(x_n)_{n \in \mathbb{N}}$ in $X$. Since $X$ is a subset of $\mathbb{R}$, this Cauchy sequence converges, meaning there exists $x \in \mathbb{R}$ such that $(x_n)_{n \in \mathbb{N}} \rightarrow x$. Next, I need to prove that $x \in (-1, 1)$.

I can't figure out if these ideas are correct, please help.

Answer 1

The easiest way to see this is to observe that the map $f\colon X\to \mathbb R$ given by $f(x)=\frac{x}{1-x^2}$ is a (surjective) isometry, if $\mathbb R$ is equipped with the usual Euclidean metric. Since $\mathbb R$ complete, so is $X$.

Remark.

Also, regarding your attempt, you have to be a bit careful, since $X$ has a different metric than the metric on $(-1,1)$ inherited from $\mathbb R$, so a Cauchy sequence with respect to the metric on $X$ is a priori not necessarily a Cauchy sequence in the usual Euclidean metric. So that reasoning doesn't quite work.