1549901369-Elements_of_Real_Analysis__Denlinger_

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2.7 Cauchy Sequences 117

First show^14 that Vu, v E JR, ..jUV :::; u ; v and equality holds <¢=> u = v.

Use this to show that a< Y1 < Y2 < · · · < Yn < Xn < · · · < X2 < X1 < b.

a+b

Also, show that 'Vn EN, lxn -Ynl <

2

n - l. Use these results to cbnclude

that {xn} and {yn} converge and have the same limit. This common limit

is called the arithmetic-geometric mean of a and b.

2. 7 Cauchy Sequences

It is often possible to prove that a sequence converges without knowing its

actual limit. The monotone convergence theorem provides one tool for doing

that. The "Cauchy criterion" we are about to study is another.

Definition 2.7.1 A sequence {xn} is a Cauchy sequence if it satisfies the

"Cauchy criterion:"

The significance of the Cauchy criterion is that, unlike Definition 2.1.4, this

criterion makes no reference to a limit "L." Thus, it can prove useful when

we do not know in advance whether a sequence has a limit, or whether it is

monotone. We shall prove shortly that every Cauchy sequence converges.

Theorem 2.7.2 Every convergent sequence is a Cauchy sequence.

Proof. Suppose { Xn} converges, say Xn --t L. Let c: > 0. Then :l no E N 3

€

n ::'.': no => I Xn - LI < 2. Then

€ €

m , n ::'.': no => lxm - LI < 2 and lxn - LI < 2

=> lxm - LI + lxn - LI < €

=> lxm - LI + IL -Xnl < €

=> lxm - L + L - Xnl < € by the triangle inequality

=> lxm - Xnl < €.

Therefore, {xn} is a Cauchy sequence. •

The next t heorem should remind you of Theorem 2.2.10 for convergent

sequences. The proofs are almost the same.

Theorem 2.7.3 Every Cauchy sequence is bounded.

14. See Exercise 1.2-B.10.