Is $\Bbb{R}$ with this metric $\rho(x,y)=\left| \frac{x}{1+|x|}-\frac{y}{1+|y|} \right|$ complete?
It is known that $$\rho(x,y)=\left| \frac{x}{1+|x|}-\frac{y}{1+|y|} \right|$$ is a metric on $\Bbb{R}$. However, I'm thinking that $\Bbb{R}$ with this metric $\rho(x,y)$ is not complete but I can't think of an example. Anyone to help?
No!
Hint: Use the sequence $x_n=(n)$
The sequence $x_n$ is Cauchy under $d$ is left as an exercise to you!
Suppose $x_n \to a$ under $d$. Then $$0=\lim \Bigg\vert \frac{n}{1+n}-\frac{a}{1+\vert a \vert} \Bigg\vert= \Bigg \vert1-\frac{a}{1+\vert a \vert} \Bigg\vert$$ and the latter has no solution. Thus $x_n$ is not convergent under $d$!