How to show the space is totally bounded? the space the Real line with bounded metric (i.e. d/(1+d), d: euclidean)
we know that totally boundedness means that there exists a finite epsilon-net.
we first approached to question by directly try to find finitely many points s.t. their metric balls cover the space. then, tried , by contradiction, but failure.

HINT: Let $d_b$ be the bounded metric, so that $$d_b(x,y)=\frac{|x-y|}{1+|x-y|}$$ for all $x,y\in\Bbb R$. Suppose that $m,n\in\Bbb Z$ and $m\ne n$; then $d_b(m,n)\ge\ldots\;$?
