1549901369-Elements_of_Real_Analysis__Denlinger_

76 Chapter 2 • Sequences

Moreover, \In EN, an :'.": u =inf A. Thus, {an} is a sequence of elements of A

such that \In E N,

1 1

u--<u<a n - n <u+-n' so

1

fan -uf < -.

n

Then, by the second squeeze principle, an -> u.

(b) Exercise 3. •

The following theorem is of fundamental importance, and will be used

frequently throughout the remainder of the course. However, its proof is easy

and is left as an exercise.

Theorem 2.3.6 (Denseness of the Rationals and Irrationals in~) Let

x be any real number. Then

(a) 3 sequence {rn} of rational numbers different from x 3 rn-> x.

(b) 3 sequence {zn} of irrational numbers different from x 3 Zn-> x.

Proof. Exercise 4. •

Theorem 2.3. 7 If faf < 1, then lim an = 0.

n->oo

1 1

Proof.^8 Suppose f af < l. Then ~ > l. Let 8 = ~ - 1. Then 8 > 0, and

1

~ = 8+1, so

1

faf=-,.

l+u

Then, by Bernoulli's inequality (see Exercise 1.3.l ')

~-. /

~l , __ + 8)n - 1 :'.": 1 + n8 /> 1 ~ so

fain = (1 + fi)n < n8 -> 0.

Therefore, by the second squeeze principle, f a f n -> 0. •

8. For an easier proof of Theorem 2.3.7, See Exercise 2.5.16.