(a) Show that (an) is bounded and divergent

1. hw 3 All the sequences are sequences of real numbers.

(1) A function f : N → N is increasing if f (i) < f (j) for any i < j. Suppose f : N → N is increasing. Use mathematical induction to prove that f (n) ≥ n for all n ≥ 1.

(2) A convergent sequence is always bounded and a bounded sequence may not always be convergent. Here is an example. Define a sequence (a_n) by

a_n= (−1)ⁿ⁻¹, n ≥ 1.

(a) Show that (a_n) is bounded and divergent.

(b) Let (σn) the sequence defined by

σn= a1+ a2+ · · · + an

n , n ≥ 1.

Show that the sequence (σn) is convergent and find its limit.

(3) Let (a_n) be the sequence defined by a_n = √

n for n ≥ 1. Show that (a_n) is not a Cauchy sequence but for each  > 0, there exists N ∈ N so that |an+1− a_n| < , whenever n ≥ N.

(4) Suppose (a_n) is a sequence. Define two new sequences (b_n) and (c_n) as follows. For each n ≥ 1, define bn = a2n−1 and cn= a2n. Suppose lim

n→∞bn = lim

n→∞cn = a. Show that (a_n) is convergent to a.

Remark. We usually denote (b_n) by (a_2n−1) and (c_n) by (a_2n).

(5) Define a sequence (a_n) by

a_n+1 = 2 + a_n 1 + an

, n ≥ 1 with a1 = 1. Prove that

(a) (a_2n−1) is increasing, and that (b) (a2n) is decreasing, and that

(c) both sequences (a2n−1) and (a2n) are convergent and that

(d) (a_n) is convergent use exercise (4) and find its limit using the properties of limit.

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