From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: analysis question
Date: 7 Dec 1999 22:06:11 -0500
Newsgroups: sci.math
Keywords: more or less equivalent statements of the completeness of the reals

In article <m33dtei4vg.fsf@localhost.localdomain>,
Larry Mintz  <kabir@citenet.net> wrote:
:
:Is the Completeness Theorem which states that  
:Every   Cauchy sequence of real numbers has a unique limit 
:
:and Bolzano-Weirstrass Theorem  which states 
:   An infintely bounded sequence of real numbers has a limit point 
:
:stating the same thing ?
:
:To me they seem verrry close in meaning.
:Larry<kabir@citenet.net>

In a way, yes. In the theory of Archimedean ordered fields, they are both
"statements of completeness": each of them can be deduced from the
supremum property, and each of them implies the supremum property. (It is
too late now for me to speculate where Archimedean property can be dropped
as a background assumption.) 

Other "statements of completeness":

every monotone bounded sequence has a limit,

every absolutely convergent series is convergent,

every continuous function on [0,1] is bounded,

every continuous function on [0,1] attains its maximum,

every continuous function on [0,1] has the Intermediate Value Property,

every continuous function on [0,1] is uniformly continuous,

[0,1] is a compact set,

[0,1] is a connected set,

variants of the above with any interval [a, b] where a<b,

or without defining connectedness: the only subsets of the field which are
both open and closed are the empty set and the whole field,

every Dedekind cut is already defined by a member of the field as a
dividing point,

Rolle's Theorem,

Mean Value Theorem,

Taylor's Theorem with remainder,

if a function has a positive derivative on (0,1) then it is increasing
  there,

every monotone function on (0,1) has a left limit and a right limit at
   every point of (0,1)


and it's a good exercise to question other Analysis theorems along these
lines.

What is not a completeness statement:

every polynomial of odd degree with coefficients in the given field has a
   root in that field,

every sequence has a monotone subsequence,

if f, defined on (0,1) has a maximum at a point c in (0,1) and if f'(c)
  exists then f'(c)=0

(why? Find an incomplete subfield of R where such a statement still
holds.)

Have fun, ZVK(Slavek).