Suppose M is a manifold. $f=(f_1, f_2): M\rightarrow \mathbb{R}\times\mathbb{R}$. Give an example that $f$ is a proper mapping but neither $f_1$ nor $f_2$ is.
For Hausdorff space $X$ and $Y$, $f: X\rightarrow Y$ is continuous. The mapping is proper means for every compact set $K\subset Y, f^{-1}(K)$ is compact in X.
I have learned that if $f_1$ is proper, then $f$ is also proper. The example asked in some sense tells us the reverse does not hold. Any ideas about how to construct the example?