1549901369-Elements_of_Real_Analysis__Denlinger_

2.7 Cauchy Sequences 119

c

Proof. Suppose that {xn} is a sequence 3 \fn E N, lxn+l - Xnl < 2n

Then, whenever m > n in N,

lxm - Xnl = l(xn+l - Xn) + (Xn+2 - Xn+1) + · · · + (xm - Xm-dl

::; lxn+l - Xnl + lxn+2 - Xn+1I + · · · + lxm - Xm-1)1

c c c

< - -+--+···+--2n 2n+l 2m-l

= 2_ 2n (1 + ~ 2 + ... + 2m-n-l 1 )

< i (2) = /:-1. (14)

Let c > 0. Since

2

:!_ 1 --> 0, 3 no E N 3 n 2:: no ::::}

2

:!_ 1 < c. Thus, from

(14), m, n 2:: no ::::} lxm - xnl < c. This means {xn} is a Cauchy sequence. •

00 1
*Example 2. 7.6 (An Application) The series LI converges. That is, the
n=l n.

sequence {t, ~!} ~~' convecges.

n 1

Proof. \fn EN, let Sn= L k!. Then, \fn EN,

k=l

l

n+l 1 n 1 I 1 1

ISn+l - Sn!= L k! - L k! = (n + 1)! ::; 2n ·

k=l k=l

(By Exercise 1.3. 13 , 2n::; (n + 1)!) Hence, by Theorem 2.7.5, {Sn} con-
verges. D

EXERCISE SET 2. 7

1. Prove directly from Definition 2.7.1 that each of the following is a Cauchy
   sequence.

(c) {n2:1} {

n-2}

(d) 3n + 4 ·

2. Prove directly from Definition 2.7.1 that each of the following is not a
   Cauchy sequence.