Let an be a sequence of real numbers It is bounded if the se

Let (a_n) be a sequence of real numbers. It is bounded if the set A = {a_1, a_2, ....} is bounded. The limit supremum, or limsup, of a bounded sequence (a_n) as n rightarrow infinity is lim_n righarrow sup a_n = lim_n rightarrow infinity (sup k lessthanorequalto n a_k) Why does the lim sup exist? If sup {a_n} = infinity, how should we define lim_n rightarrow infinity sup a_n? If lim_n rightarrow infinity a_n = - infinity, how should we define lim sup a_n?

Solution

a> the lim sup always exists as the set A has real numbers as its elements and it may be the limit doesn\'t exists but the lim supremum will alays exist as real number.

and the limsum will be the same for all the a values within the set A.