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I am going through a self teaching journey in mathematics. Right now I am reading Book of Proof, by Richard Hammack, and on the Chapter on Counting, I came across the following exercise:

Given a sphere $S$, a great circle of $S$ is the intersection of $S$ with a plane through its center. Every great circle divides $S$ into two parts. A hemisphere is the union of the great circle and one of these two parts. Show that if five points are placed arbitrarily on $S$, then there is a hemisphere that contains four of them.

$\bf{My\,answer}$: Let $A=\{p_1,p_2,\ldots,p_5\}$ be a set of arbitrarily placed points on the sphere $S$.

Since 3 points are sufficient to define a plane in $\mathbb{R}^3$, we can construct a plane $P$ by taking any two elements of $A$ and the center $c$ of $S$.

Without loss of generality, choose the points $p_1,p_2$ and $c$. The intersection of the plane $P$ with the sphere $S$ is, by definition, a great circle of $S$ and it defines two hemispheres $H_1$ and $H_2$.

Since $p_1$ and $p_2$ are on a great circle of $S$, both hemispheres $H_1$ and $H_2$ contain $p_1$ and $p_2$.

Now, let's look at the remaining points $p_3,p_4$ and $p_5$. Notice that for $k\in\{3,4,5\}$, it follows that $p_k\in H_1$ or $p_k\in H_2$. Since we have three points to place in two different regions, by the Pigeonhole Principle, it follows that one of $H_1$ or $H_2$ contain at least two of $p_3,p_4,p_5$. Without loss of generality, assume $H_1$ contains $p_3$ and $p_4$. By construction, $H_1$ also contains $p_1$ and $p_2$. Thus, there is a hemisphere of $S$ that contains four of the five arbitrarily placed points on $S$.

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I desperately need some feedback on my proofwriting skills. It feels like my argument is solid, but at the same time, my writing might be somewhat convoluted. (I might have defined something in non standard ways, of even wrong ways)

It might be important to note that English is not my native language.

Feel free to be very critical. I wanna be able to write proofs acceptably.

N. F. Taussig
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Igor Mello
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Your proof is correct, i recommend to you trying to simpliciate your arguments un search to elegance. Something like this:

If you pick two different points out of your 5 (if the 5 points are the same the statement is clearly true), these two points define at least one great circle. Nos pick one ofrece the great circles they define. This great circle divides the sphere into two hemispheres. Now apply the pigeonhole principle to the other three points between these two hemispheres.

Anyway, your proof is perfect!

Guillermo García Sáez
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