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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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    Yes that makes sense but I'm still not sure how to define f. We can suppose the rectangle is [a,b]x[a,b]2012-03-13
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    Hint: take the minimum of a certain function of $x$ and a certain function of $y$.2012-03-13
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    I'm sorry I'm still stuck. Any additional help would be appreciate.2012-03-13
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    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?2012-03-13
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    Would the parabola f(x)=-(x-a)(x-b) work?2012-03-13
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    Yes, it would work2012-03-14
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    let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/2773/discussion-between-caligurl11-and-robert-israel)2012-03-14
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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