product of two continuous proper maps

Question

Let $f,g: \mathbb{R} \to \mathbb{R}$ be continuous proper maps (i.e., $f^{-1}(K)$ is compact for all $K \subset \mathbb{R}$ compact). Is $fg$ proper?

Answer 1

Yes, the product of two proper maps $f,g\colon\mathbb{R}\to\mathbb{R}$ is proper.

Let $K$ be compact, and $N = \max \{ \lvert x\rvert : x \in K\}$. Since $f$ is proper, $f^{-1}([-(N+1),N+1])$ is compact, so there is an $N_f > 0$ such that $f^{-1}([-(N+1),N+1]) \subset [-N_f,N_f]$. Similarly, there is an $N_g > 0$ such that $g^{-1}([-(N+1),N+1]) \subset [-N_g,N_g]$.

For $\lvert x\rvert > M := \max \{ N_f, N_g\}$, we have $\lvert f(x)g(x)\rvert > (N+1)^2 > N$, so $(fg)^{-1}(K) \subset [-M,M]$ is bounded. Since $fg$ is continuous, $(fg)^{-1}(K)$ is also closed. By the Heine-Borel theorem, $(fg)^{-1}(K)$ is compact, hence $fg$ is proper.