Let X be a set, and define d(x,y) ={0, if x=y; 1, otherwise A sequence (xn) ⊆ X is called eventually constant if there is N ∈ N such that xm=xn; m, n > N. Show that any eventually constant sequence converges.

4. Let (sn)-1 be a sequence of real numbers. Prove that lim sup |sn| <∞ if and only if the sequence (sn) is bounded.

Let b1, b2, b3,... be the sequence of numbers defined by b1 = 1, and b, V6 + bn-1 if n > 2. Prove that 0 1.

12.B. Let 2i be any real number satisfying zı > 1 and let æ2 .., Xn+1 is monotone and bounded. What is the limit of this sequence? 12.C. Let r1 2 - 1/T1, (Xn) 2 - 1/Tn, . Using induction, show that the sequence X ... 1, x2 = V2+ x1 , . . ., Xn+1 V2 + xn, .... Show that the sequence (xm) is monotone increasing and bounded. What is the limit of this sequence?

Suppose (an) (b), and (care sequences such that (a converges to A, (b) converges to A, and a, c, ≤ b, for all n. Prove that (c)-1 converges to A.

Show that the infinite sequence 5. an = √ (n2 + 1) − √ (n2 − 1) (n ≥ 1) converges by showing that it is monotone and bounded. You do not need to find the limit of the sequence

Let {n} and {n} be real sequences. Show that if xn → x and yn →y then for all a, ß ER, axn + Byn → ax + By