Suppose $f \in \mathbb{R}[x]$ and define $g \colon \mathbb{R} \to \mathbb{R}$ by $g(x) = \frac{f(x)^2}{(x^2+1)^{d+1}}, \text{where } d = \deg(f)$
I'm looking for a quick proof as to why $g$ is bounded above and Lipschitz.
Edit: $g$ is not proper as mentioned below.