What is your definition of limsup. The definition I know is exactly lim b_m, but it seems like you were given a different one to start with, and are asked to show that the two are equivalent.

In part b), you wrote "sup an <= sup{a_m,a_m+1,...}" which is where you got stuck. You're right to get stuck there, because that's not necessarily true. But that's because L is not sup an, it's limsup an, which has a different definition