O'Neill's Elementary Differential Geometry poses the following problem (exercise 3.14 of chapter 4): Prove every patch $x:D\rightarrow M$ in a surface M is proper. (Hint: Use exercise 4.3.13. Note that $(x^{-1}y)y^{-1}$ is continuous and agrees with $x^{-1}$ on an open set in x(D)).
Exercise 4.3.13 is at exercise 4.3.13
Consider a point p in x(D). $(x^{-1}y)y^{-1}=x^{-1}(p)$ in a neighborhood of p.
$\lim_{q\to p} x^{-1}(q)=x^{-1}(p)$. $x^{-1} is continuous, so x is a proper patch.