If an greaterthanorequalto 0 and bn greaterthanorequalto 0 p

If a_n greaterthanorequalto 0 and b_n greaterthanorequalto 0, prove that lim sup a_n b_n lessthanorequalto (lim sup a_n) (lim sup b_n).

Solution

Assume that a_n 0 and b_n 0
Let
x_n := sup{a_n, a_n+1, . . .} , y_n := sup{b_n, b_n+1, . . .}, and z_n = sup{a_nb_n, a_n+1b_n+1, . . .}

Since a_kb_k x_ny_n for all k n,

We have z_n x_n y_n and consequently ( bounded)

lim sup(a_nb_n) = limz_n lim(x_ny_n) =( limx_n · lim y_n) ( lim sup an ) ( lim sup bn.)

Hence,

lim sup(a_nb_n) ( lim sup an ) ( lim sup bn.)