Let {an} be a bounded sequence of numbers in r. we define a new sequence bn} as follows sup{an n n} = 1. prove that {by} is monotone. definition. if {an} is a bounded sequence in r, then the limsup, denoted lim sup an, is defined as: n oo lim sup an n > n} lim sup an (1) noo n oo 2. prove that the limit in (1) exists. 3. based on the definition of the lim sup above, what would the definition of lim inf be? 4. using the definition you gave in the previous problem, show that for bounded sequences, the lim inf always exists 5. prove that if lm sup{an} lim inffan} l. then the limit of {an} exists and equals l