Suppose that {xn} and {yn} are real sequences. a) Prove that provided that none of these sums is of the form - . b)

Suppose that {xn} and {yn} are real sequences.
a) Prove that

provided that none of these sums is of the form ˆž - ˆž.
b) Show that if limn†’ˆž xn exists, then

and

c) Show by examples that each of the inequalities in part (a) can be strict.

lim inf xn + lim inf yn lim inn) S lim sup x lim inf y S m sup(n t yn) lim sup+lim sup yn lim inf(x" + yn) = lim xn + lim infy" lim sup(xn+yn) = lim xn +lin supyn