Hello. I read the textbook and found that common eigenfunctions are even possible for degenerate eigenvalues. Let's say operators A and B commutes and eigenvalue a of operator A is N-fold degenerate, means that there are N linearly independent eigenfunctions having same eigenvalue a. These eigenfunctions are not necessarily common eigenfunctions of operator B, unlikely to non-degenerate eigenfunctions that are always common for commuting operators. However, the textbook said that we can always find a set of new N linearly independent eigenfunctions of eigenvalue a that are common for operator B by linear combination of old eigenfunctions of a. It is really fantastic theorem but I couldn't find its proof. Could you please give me the proof of this? If I fully understand the proof, then I will be really confident that commuting operators have common eigenfunctions for all their eigenvalues, even some eigenvalues are degenerate.