Re: (.999999... == 1) = 1 <--No, if it does 0 =1

From: Dave Seaman (dseaman_at_no.such.host)
Date: 10/03/04 

Date: Sun, 3 Oct 2004 13:24:53 +0000 (UTC)

On 03 Oct 2004 11:58:04 GMT, S. Enterprize Company wrote:

> You missed the whole point. I showed something ridiculous with 1 = 0, to show
> that .99999... = 1 is ridiculous.

The notation 0.999... is naturally associated with the Cauchy sequence

        9/10, 99/100, 999/1000, ... (1)

while 1.000... is associated with the Cauchy sequence

        1, 1, 1, 1, 1, .... (2)

Each of (1) and (2) is a Cauchy sequence of rationals. Clearly, they are
not the same sequence. However, we can define an equivalence relation on
such sequences.

        Definition. We say that { a_k } ~ { b_k } if, for every
        epsilon > 0, there exists N > 0 such that | a_k - b_k | <
        epsilon for every k > N. (3)

The relation ~ is reflexive, symmetric and transitive, which is what it
means to be an equivalence relation. An equivalence relation on a set D
partitions the set into a collection of sets known as equivalence classes
with an important property. Writing [x] to denote the equivalence class
of x with respect to ~, we have that for all x, y in D, x ~ y iff [x] =
[y], meaning that x and y belong to the same equivalence class.

This technique is used to define the integers as equivalence classes of
pairs of natural numbers with the equivalence relation (a,b) = (c,d) iff
a+d = b+c. Thus the integer -1 is identified with the equivalence class
that includes such pairs as (0,1), (5,6), and (137,138).

Similarly, the rationals are equivalence classes of pairs in Z x (Z\{0})
with the property that (a,b) ~ (c,d) iff a*d = b*c. Thus, the rational
number 1/2 is identified with the equivalence class that includes such
pairs as (1,2), (3,6), and (47,94). Obviously 1/2 and 47/94 are not the
same fraction, but they represent the same rational number.

Following the same technique, the real numbers can be defined as
equivalence classes of rationals with the equivalence relation (3).
According to this definition, the Cauchy sequences (1) and (2) are
members of the same equivalence class and therefore represent the same
real number. That is, the real number 1 is an equivalence class that
includes sequences (1) and (2) above. Thus, (1) and (2) are not the same
sequence, but they represent the same real number.

This is what is meant by the equation 0.999... = 1. 

-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>

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