1549901369-Elements_of_Real_Analysis__Denlinger_

2.7 Cauchy Sequences 123

*PROPERTIES EQUIVALENT TO COMPLETENESS

Several of the major theorems we have encountered so far are equivalent

to the completeness property. We h ave derived them from the completeness

property. We shall now show that in an Archimedean ordered field, any one of

them will yield the completeness property as a consequence. The equivalences of

these ideas is a result of pivotal significance in the foundations of real a nalysis.

*Theorem 2. 7. 7 Let F denote an Archimedean ordered field. The following

conditions are equivalent:

(a) F is complete;

(b) Every bounded monotone sequence in F converges in F ;

( c) Cantor's Nested Intervals Theorem;

(d) The Balzano-Weierstrass Theorem (every bounded sequence has a conver-

gent subsequence};

(e) Every Cauchy sequence in F converges in F.

Proof.
(a) =? (b). Monotone convergen ce theorem (Theorem 2.5.3).
(b) =? (c). Cantor's nested intervals theorem (Theorem 2.5.17).
(c) =? (d). Balzano-Weierstrass theorem (Theorem 2.6.16).
(d) =? (e). Theorem 2.7.4.
(e) =? (a). Proof: Suppose F is an Archimedean ordered field in which
every Cauchy sequence converges. Let S be a nonempty subset of F with an
upper bound. Then 3 B ~ F 3 \:/x E S, x :S: B. We sh all prove that S has a
least upper bound in F. We begin by constructing a sequence { Xn} as follows:

Step 1. By the Archimedean property, the set

A= {n EN: n is an upper bound for S}

is nonempty. Hence, by the well- ordering property, 3 x 1 = min A.

Then x 1 is an upper bound for S, but x 1 - 1 is not.

Define x

2

= { x 1 - ~ if x 1 - ~is an upper bound for S,

x 1 otherwise.

Then,