# Find the completion of an incomplete metric space?

I have the incomplete metric space $M = (\mathbb{R}, d)$, where the metric $d$ is given by $d=|\tanh(x)-\tanh(y)|$.

I have the following definition of a completion:

"Given an incomplete metric space $(X, d)$, a completion is a pair $((X', d'), \Phi)$, where $(X', d')$ is a complete metric space and $\Phi$ is an isometry between $(X, d)$ and $(X', d')$, sucht that $\Phi(x)$ is dense in $X$."

I know already that ${a_n}=n$ is a Cauchy sequence in $M$, but does not converge in $M$, since $\infty \notin \mathbb{R}$ (hence $M$ is not complete).

So how exactly can I find the completion of $M$?

• Consider $\tanh \colon (\mathbb{R},d) \to (\mathbb{R}, \lvert\,\cdot\,\rvert)$. What do you know about that map? – Daniel Fischer Nov 11 '17 at 14:39
• For a general strategy that fits well with the answer of @HagenvonEitzen, see math.stackexchange.com/questions/2469018/… – Lee Mosher Nov 11 '17 at 14:46

Instead of adding points to $\Bbb R$, simply note that $x\mapsto \tanh x$ is by definition an isometry $(\Bbb R,d)\to ((-1,1),d_0)$ where $d_0$ is the standard metric, $d_0(x,y)=|x-y|$. This suggests that $(([-1,1],d_0),\tanh)$ is a good candidate ...