Denote $ d_1(x,y)=|x-y|\qquad d(x,y)=\frac{d(x,y)}{1+d(x,y)} $ You should check that $(\mathbb{R},d)$ and $(\mathbb{R},d_1)$ are metric spaces. Moreover, from the point of view of theory of metric spaces, they are essentially the same! Indeed, consider maps $ f:(\mathbb{R},d)\to(\mathbb{R},d_1):x\mapsto x\qquad g:(\mathbb{R},d_1)\to(\mathbb{R},d):x\mapsto x $ You should check that they are continuous. Then we easily see that $f(g(x))=x$ and $g(f(x))=x$, so this maps are inverse to each other. Hence $f$ and $g$ are isomorphisms between metric spaces $(\mathbb{R},d)$ and $(\mathbb{R},d_1)$.
For isomorphic spaces notion of compactness coinside (but not completeness!). Recall what you know about compactness/non-compactnes of $\mathbb{R}$ with standard metric.