7. Let {bk} be a real sequence and bE R.

(a) Suppose that there is an N E N such that I b - bk I :::; M for all k 2: N. Prove that n N nb - 2: bk :::; 2: Ibk - bl + M(n - N)

k=l k=l for all n > N.

(b) Prove that if bk ~ bask ~ 00, then as n ~ 00.

(c) Show that the converse of

(b) is false.