Is C(R) compact space in regard to given metric

Question

Let $C(\mathbb{R})$ be a set of continous functions $f:\mathbb{R}\to\mathbb{R}$ and $d:C(\mathbb{R})\times C(\mathbb{R})\to \mathbb{R}$ defined as $$d(f,g)=\begin{cases} 0 , & f=g ;\\ \sup\limits_{x\in\mathbb{R}} (\arctan{ |f(x)|} + \arctan{ |g(x)|}) , & f\ne g . \end{cases}$$ I proved that this is indeed a complete metric space by checking that every Cauchy sequence converges, but I would like to know if it is compact metric space, because then I could directly deduce completeness? Any help is welcome. Thanks in advance.