# Polygonal line connecting z to infinity intersects boundary of rectangle

Show that a polygonal line $\gamma$ connecting $z$ to infinity intersects the boundary of every rectangle $R$ containing $z.$

So we want to consider $t_0 = \sup \{t : \gamma(t) \in R\}$. This seems intuitive but I'm not exactly sure how to put it in words. Also the intermediate value theorem might be helpful.

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Hint: find a continuous function $f$ that is positive inside your rectangle and negative outside (or vice versa if you prefer), and use the Intermediate Value Theorem on $f(\gamma(t))$.
Hint: take the minimum of a certain function of $x$ and a certain function of $y$. – Robert Israel Mar 13 '12 at 19:07
If the rectangle goes from $x=a$ to $x=b$, what's a function of $x$ that is positive when $x$ is between $a$ and $b$, and negative when $x$ is outside that interval? – Robert Israel Mar 13 '12 at 23:20