Question

define these please!

following definitions: . A subsequential limit of (sn) is. . . , an or lim sup snSu p . lim inf sn-, , . In

Answer 1

A subsequatial limit of (S_n) is nothing but the limit of some subsequence of (s_n) lim_n rightarrow infinity snp s_n = The supermum of the set of all sub sequential limits of (s_n) lim_n rightarrow infinity inf s_n = The infimum of the set of all subsequential limits of (s_n) Example Consider (s_n) = (- 1)^n = { - 1, 1, - 1, 1.....} Now if n = even then n = 2k (for some k elementof kz) Then a subsequence of (s_n) is (s_2k) = (- 1)^2k = {1, 1,.....} so we can say 1 is a subsquential unit of (s_n) Similar if n = odd then n = 2k + 1 Then a subsequence of (s_n) is (s_24 + 1) = (- 1)^24 + 1 = { - 1, - 1.....} S we can also say = 1 is also a subsequential limit of (S_n) (S_n) does not have any other subsequence and hence no other subsequential limit Now lim_n rightarrow infinity sup (s_n) = sin p {1, - 1} = 1 (Bus the def^n) and lim_n