Completion of $\mathbb{R}$

Question

Is it possible to define a metric on $\mathbb{R}$ such that $\mathbb{R}$ is not complete? I know it for this to happen, we would need to construct a Cauchy sequence that does not converge in $\mathbb{R}$, but I'm having trouble doing this. Would the completion of $\mathbb{R}$, with respect to this metric, be some subset of the complex numbers?

Answer 1

Let $f(x)=\arctan x$, define $d(x,y)=|\arctan x - \arctan y|$ then $d$ is a metric since it is trivially positive definite, symmetric and sub-additive. Now can you show it is not complete?

Hint: consider the sequence $1,2,3,4,\cdots$ under this metric.