7. [Extra Credit, 5pt] A set of numbers A is eventually periodic if there is a number p (the period) and some number n so that m is in A if and only if m+p in A, for all m 2 n. E.g., 1, 5, 10, 11, 20, 21, 30, 31, 40, 41, ... is eventually periodic with period 10 starting at n = 10. Show that a language Lover the single-letter alphabet {a} is regular if and only if the set of lengths of strings in L: A = {lwl: w in L} is eventually periodic. 7. [Extra Credit, 5pt] A set of numbers A is eventually periodic if there is a number p (the period) and some number n so that m is in A if and only if m+p in A, for all m 2 n. E.g., 1, 5, 10, 11, 20, 21, 30, 31, 40, 41, ... is eventually periodic with period 10 starting at n = 10. Show that a language Lover the single-letter alphabet {a} is regular if and only if the set of lengths of strings in L: A = {lwl: w in L} is eventually periodic