1. Homework 1 A sequence (an

1. Homework 1

A sequence (an) of real numbers is convergent to a ∈ R if for every  > 0, there exists N > 0 such that |an− a| <  whenever n ≥ N_.

A sequence (an) of real numbers is a Cauchy sequence if for every  > 0, there exists N_ > 0 such that |a_n− a_m| <  whenever n, m ≥ N_.

(1) Use definition to show that

n→∞lim

3n²− n − 1 5n²+ 4n + 3 = 3

5.

(2) Let (a_n) be a sequence of real numbers and K > 0. Suppose that

|a_n+1− a_n| < K

2ⁿ, n ≥ 1.

Show that (an) is a Cauchy sequence.

(3) Let (a_n) be a sequence of integers. Suppose that (a_n) is convergent. Show that there exists N > 0 so that an= aN for all n ≥ N.

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

(a) lim

n→∞r_n= 0,

(b) 0 < |pn− αq_n| < r_n, for n ≥ 1.

Show that α must be an irrational number. (Hint: use (3)).

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