The question is taken from Cool Induction Problems.
Quoting:
(**) A sphere is covered with some number of “caps” which are hemispheres. Prove that it is possible to choose four hemispheres, and remove all others, while still keeping the sphere covered. (Hint: Sometimes it is easier to prove a more general statement than the one given.)
How should I proceed with induction? I tried assuming that the statement is true for $k$ and proving that it is true for $k+1$, but this requires proving an another strong statement that would finish the proof: For $k+1$ caps covering the sphere, you can find $k$ that also cover the sphere for $k\geq 4$. How should I work on proving this statement, or should I work on a different statement?
Also, how does this generalize to $n$-spheres, and even more generally, arbitrary sets of points? (Some places mentioned Helly’s theorem.)
Thank you in advance.