Is the theorem correct (is it a true mathmatical statement) and is the proof correct? Please justify/explain.

I think that the theorem is correct but there is something wrong with the proof. Can this be explained more clearly?

8. (10 points) Consider the language L (aa)mc(bb) m 30 This lan guage contains the strings c, aacbb, aaaacbb, et cetera. Consider the following theorem and proof Theorem: Lis not regular. Proof: We show that in the regular expression game, A (the opponent) can always win. Suppose A picks the integer n 50, B picks any string (aa) c(bb)" of length larger than 50, then if k 1 A picks z e, y aa, z (aa) -1c Now whatever value of i B picks, the string ryiz is in c(bb)". L because ryiz is (aa) (aa) -1c(bb)m. If k 0 then (because the string has length larger than 50), m 2 and A picks C, y bb, and z 00 m-1 1. This is possible because m 2. Whatever value of i B picks. the string Taiz is in L because ayiz is C (bb) m-1 Therefore the opponent (A) can always 1. win, so L is not regular (a) Is the theorem correct? That is, is it true as a mathematical statement? Justify your answer (b) Is the proof correct? Justify your