I'm trying to solve the following question:
Find a metric $d$ on $\mathbb{R}$ that is equivalent to the usual metric and has the property that the sequence $(n)_{n=1}^{\infty}$ is a Cauchy sequence.
I know that if I have two metric spaces $(M, \rho)$ and $(N, \sigma)$ and a 1-1 function $f : M \to N$ such that $f$ is a homeomorphism of $M$ onto $f(M)$, then the topologies generated by the two metrics $\rho$ and $\sigma$ are equal. So if $\tau$ is the usual metric on $\mathbb{R}$, then I need to find a metric $\rho$ on $\mathbb{R}$ which has the property that the sequence $(n)_{n=1}^{\infty}$ is Cauchy, and such that there is a homeomorphism of $f : \mathbb{R} \to \mathbb{R}$ of $(\mathbb{R}, \tau)$ onto $(f(\mathbb{R}), \rho)$.
One of the hints I have is to find a homeomorphism of $\mathbb{R}$ onto a bounded interval, but I'm not yet sure how to proceed.
Any ideas or answers are greatly appreciated!