=+4.2. 1

(a) Prove that

(lim sup An ) 0 (lim sup Bn ) > lim sup ( A, B,),

(lim sup An ) U(lim sup B.) = lim sup ( A, B,),

(lim inf A„) (lim inf B.) = lim inf (A,B ,, ),

(lim inf A„) U(lim inf B ,, ) C lim inf (A ,, UB ,, ).

Show by example that the two inclusions can be strict.

(b) The numerical analogue of the first of the relations in part

(a) is

(lim sup.x„) ^ (lim supyn) ≥ lim sup (x, Ay,).

Write out and verify the numerical analogues of the others.