# 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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## 1 Answer

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] –  caligurl11 Mar 13 '12 at 18:32
Hint: take the minimum of a certain function of $x$ and a certain function of $y$. –  Robert Israel Mar 13 '12 at 19:07
I'm sorry I'm still stuck. Any additional help would be appreciate. –  caligurl11 Mar 13 '12 at 23:10
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
Would the parabola f(x)=-(x-a)(x-b) work? –  caligurl11 Mar 13 '12 at 23:32
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