Metrics w.r.t which $\mathbb R$ is not complete

Question

I have seen some examples of metrics on $\mathbb R$ which make it incomplete, like $d(x,y)=|e^{-x}-e^{-y}|$ and $d(x,y)=\arctan x - \arctan y$ where $1,2,3,\ldots$ is a Cauchy sequence which doesn't converge. I am curious about what other examples there are. Is it always infinity which turns out to be 'missing' or are there metrics with Cauchy sequences that do not converge but remain bounded?

Answer 1

Cauchy sequences always remain bounded:

If $\epsilon = 1$, there is $N$ such that for all $n,m>N$ $$d(x_n,x_m)<1$$

Therefore, $d(x_n,0)\leq d(0,x_{N+1})+d(x_{N+1},x_n)<d(0,x_{N+1})+1$, for $n>N$ and a fixed. This means that the whole sequence is inside the ball with center $0$ and radius $\max(d(0,x_{N+1})+1, d(x_1,0),d(x_1,0),...,d(x_N,0))$.

In your examples, $1,2,3,...$ is bounded.

Answer 2

That depends on how much leeway one has in specifying the metric. Consider for example the fact that there exists a bijective map $\phi\colon\mathbb R \to \mathbb R \setminus \mathbb Q$. If you define the metric on $\mathbb R$ by setting $d(x,y)=|\phi(x)-\phi(y)|$, it will be a metric alright, and clearly not complete.

Answer 3

If you restrict yourself to a bounded part of the reals,so some closed interval $[A,B]$ then all equivalent metrics on that part are defined on a compact space, so are always complete.

This is the reason that an equivalent non-complete metric on $\mathbb{R}$ has its non-completeness at infinity, as it were.