Tell if a set $K \subseteq\mathbb R $, closed in $\tau_d$ and bounded by the distance d is necessarily compact in $\tau_d$

Question

Considering the following metric over $\mathbb R$:

$d(x,y)=|x-y|/(1+|x-y|)$

I have to

1) find if ($\mathbb R$,d) is a complete metric space

2)Tell if a set $K \subseteq\mathbb R $, closed in $\tau_d$ and bounded by $d$ is necessarily compact in $\tau_d$

So I already proved part 1: I took a Cauchy sequence with respect to d and proved that the completeness of $\mathbb R$ with the usual topology implies the sequence is convergent and therefore $(\mathbb R,d)$ is a complete metric space . How do I proceed with part 2 ?

Note: the answer is No!, but I'd like to know why

Answer 1

If $B_d(x_0,\epsilon )$ is the ball that it's center is $x_0$ and it's radius is $\epsilon $ with the metric $d$. then

$$B_d(x_0,\epsilon )=\{ x: d(x,x_0)<\epsilon\}=\{x: \frac{|x-x_0|}{1+|x-x_0|}<\epsilon \}=\{x: 1+|x-x_0|<\frac{1}{1-\epsilon } \}=\{ x: |x-x_0|<\frac{\epsilon }{1-\epsilon} \}=B(x_0, \frac{\epsilon}{1-\epsilon}) $$

So every open set in $(\mathbb{R},d)$ is an open set in $\mathbb{R}$.