Show that the sequence is monotone and bounded

1. hw 6 (1) Let x1∈ R with x1> 1. Define (xn) by

xn+1= 2 − 1

x_n for n ∈ N.

Show that the sequence is monotone and bounded. What is its limit?

(2) Let (an) be a sequence in R^p and a0∈ R^p. Assume that there exists 0 < ρ < 1 such that kan+1− ank ≤ ρkan− an−1k for any n ≥ 1.

Prove that (a_n) is convergent in R^p. (3) Let a ≥ 1 and 0 < z1< 1 +√

1 + 4a

2 . Define a sequence (zn) by z_n+1=√

a + z_n, for n ∈ N.

Show that (zn) is convergent and find its limit. Use two different methods to prove the convergence of (zn).

(a) Use mathematical induction to show that (zn) is increasing and bounded above. (A sequence (an) is bounded above if there exists U ∈ R so that aⁿ≤ U for all n ∈ N.) (b) Use the following observation:

zn+1− zn=√

a + zn−pa + zn−1= zn− zn−1

√a + z_n+√

a + z_n−1 and for any n ∈ N,

0 < 1

√a + zn+√

a + zn−1

≤1 2. Use exercise (2).

(4) Determine the convergence/divergence of the sequence (xn) defined by xn= 1

n + 1+ 1

n + 2+ · · · + 1 2n =

n

X

i=1

1 n + i. (5) Let (x_n) be a sequence in R^p.

(a) If (x_n) is convergent to x ∈ R^p, show that lim_n→∞kx_nk = kxk.

(b) If limn→∞kxnk = 0, show that limn→∞xn= 0.

(c) If limn→∞kxnk exists, is the sequence (xn) convergent? If yes, prove it. If not, give a counterexample.

(6) Assume that lim

n→∞(kxn+1k/kxnk) = r with r < 1.

(a) Show that there exist positive real numbers ρ and C and a natural number N with r < ρ < 1 so that

0 < kxnk < Cρⁿ for n ≥ N.

(b) Show that lim

n→∞x_n= 0.

(7) Let (a_n) be a sequence in R^p such that

n→∞lim

pkan nk = r with r > 1.

Prove that (a_n) is unbounded. (Hint: estimate ka_nk_Rpand use the Bernolli’s inequality: for x > 0, we have (1 + x)ⁿ ≥ 1 + nx for any n ∈ N.)

(8) Let (an) be a sequence in R^p and C > 0.

(a) Assume that

kan+1− ank ≤ C

n(n + 1), n ≥ 1.

Prove that (an) is convergent in R^p. (Hint: show that (an) is a Cauchy sequence in R^p and use the completeness of R^p.)

1

2

(b) Let (xn) be a sequence of real numbers defined by xn+1= xn+ (−1)ⁿ

(2n + 1)!

with x1= 1. Test the convergence/divergence of (xn). (Hint: use (a)).

(9) Let Q be the subset of all rational numbers and

d(x, y) = |x − y|, x, y ∈ Q.

Then (Q, d) is a metric space. In this exercise, you will learn that (Q, d) is not a complete metric space.

(a) For each n, define

xn = 1 + 1

1!+ · · · + 1

n!, n ≥ 1.

Prove that xn∈ Q for all n ≥ 1.

(b) Prove that (x_n) is a Cauchy sequence in (Q, d).

(c) Prove that (x_n) is divergent in (Q, d). Hint: Suppose (xn) is convergent to some x = p/q in Q. Define pn= n!x_nfor n ≥ 1. Show that |n!x−p_n| > 0 for n ≥ 1. Estimate |n!x−p_n|.

Then use the suggested exercises below to show that such x does not exist.

To do (9), you need to recall the exercise from hw 1 and hw 2 in Calculus I: (you do not need to turn in the following exercises) If you are not familiar with the exercises below, I suggest that you should solve these.

(1) Let (a_n) be a sequence of integers. Suppose that (a_n) is convergent in R. Show that there exists N such that a_n= a_N for any n ≥ N.

(2) Let α be a real number. Suppose that there exist sequence of integers (pn) and (qn) and a sequence of positive real numbers (rn) such that

(a) limn→∞rn = 0.

(b) 0 < |pn− αqn| < rn for any n ≥ 1.

Show that α must be an irrational number.