A function $f: \mathbb R \rightarrow \mathbb R$ is called proper if $f^{-1}(C)$ is compact for every compact set C. Prove or give a counterexample: If $f,g$ are continuous and proper, then the product $fg$ is proper.
My attempt: Let $h(x)=f(x)g(x)$ and product of two continuous function is continuous. This implies $h(x)$ is continuous.
Now, I claim that $h(x)$ is proper by showing that $f^{-1}(C)$ has finite number of subcovers.
Is this idea is correct? If not can anyone suggest the contradict example?