For a set $X$, let $\mathbb{R}^{X}$ be the set of all maps from $X$ to $\mathbb{R}$.
For $f,g\in\mathbb{\mathbb{R}}^{X}$, define $$d(f,g) = \sup_{x\in X}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}.$$
I am trying to show that $(\mathbb{R}^{X},d)$ is a metric space but I can't get the bounds in the right way.