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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 $f(\gamma(t))$.

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    I guess I'm still not clear on how to use the Intermediate Value Theorem on f(γ(t)).2012-03-14