What open cover should i consider in order to show $(X,d)$ is not compact?

Question

Let $X= \mathbb R$,let

$d(x,y)=\frac{\left|x-y\right|}{1+\vert x-y \vert} \forall x,y \in \mathbb R$.

Show that $(X,d)$ is not compact.

What is the form of open sets of $(X,d)$?

Answer 1

If $X$ were compact, the map $f \colon X \to \mathbb{R}$, $x \mapsto d(x,0)$ would have a compact image. Does it?

Answer 2

If you want an open cover, consider $\{B(0, 1- \frac{1}{n}): n \in \mathbb{N}^+ \}$. Every $x \in \mathbb{R}$ has distance $d(0, x) < 1$, so is covered by one of these open balls. ANd as $n$ gets larger $d(0,n) \rightarrow 1$, so we cannot have a finite subcover, as the ball with the largest radius will not contain some $n \in \mathbb{R}$.

Similarly, we can consider the sequence $x_n = n$ and show it has no convergent subsequence (despite it even being Cauchy)