Very weak version of the Banach-Tarski paradox: Let be the usual unit circle in . Remove one point from from the circle (for example ), and call this punctured unit circle . Then it is possible to find two subsets of such that after rotating to , we have .
Proof: The proof is short and easy. Identify the plane with for convencience. Let . Let (we adopt the convention that ). Then is a proper (countable) infinite subset of . Let . We have . Now, let be the rotation . Then and .
The proof is easy to follow, but the technique is conceptually the same as the technique used to prove the strong version of the paradox. The technique is basically to find some infinite subset of that is regular enough to "almost rotate into itself" - with "almost" here meaning "all but finitely many points".
Why is the above result interesting? My personal opinion is its conceptual carriage; it gives us the idea of how to prove stronger results such as the Banach Tarski-paradox. And also that mathematics is not always intuitive.
I'll end this post with a link to my own undergrad project paper about the strong Banach-Tarski paradox (contains some minor typos, by the way). CLICK HERE.