Re: Size of equivalence class of Cauchy sequences

In article <1170102825.017538.158290@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
MoeBlee <jazzmobe@xxxxxxxxxxx> wrote:

On Jan 11, 1:29 pm, "Stephen J. Herschkorn" <sjhersc...@xxxxxxxxxxxx>
wrote:
MoeBlee wrote:

So, let me see if I understand correctly: If we use the equivalence
classes of Cauchy sequences method to define real numbers, then each
real number is itself a set with cardinality that of the set of real
numbers. (?)

Yes, I think that is correct.

I know you mentioned something toward a proof, but would spell out for
me in more detail (hopefully with mainly basic set theory and not too
much real analysis, though I know that such a limitation can't be
insisted on) that each real number (equivalence class of Cauchy
sequences of rationals) has cardinality at least as great as the
cardinality of the set of real numbers?

I hope you don't mind if I give the idea I had:

It would be enough to show that there are uncountably many
Cauchy sequences of nonzero rationals that converge to 0; for given
such a null sequence, {z_n}, and given any Cauchy sequence {a_n}, then
the sequence {a_n+z_n} is also a Cauchy sequence and is equivalent to
{a_n}.

Consider the harmonic sequence H={1/n}_{n>0}

which is a Cauchy sequence of rationals that converges to 0. Any
subsequence of this is also a Cauchy sequence of rationals that
converges to 0. Given any infinite subset S of N, the set S determines
a subsequence of H by taking only those terms whose index lies in
S. Call this sequence H_S, and the terms h_{S,n}. Then H_S is a Cauchy
sequence of nonzero rationals that converges to 0, and therefore if
{a_n} is any Cauchy sequence of rationals, so is {a_n + h_{S,n} } .


If S is different from T, both infinite subsets of N, then there is a
least positive integer k such that h_{S,k} is different from h_{T,k};
then a_k+h_{S,k} is different from a_k+h_{T,k}, so the sequences
{a_n + h_{S,n}} and {a_n + h_{T,n}} are distinct (though equivalent).

Thus, we have an injection from the collection of all infinite subsets
of N to the set of Cauchy sequences of rationals that are equivalent
to {a_n}, and so the equivalence class of {a_n} contains at least
2^{aleph_0} elements.

Since the set of all sequences of rationals has cardinality
2^{aleph_0}, we already have that the cardinality of the equivalence
class is at most 2^{aleph_0}, so we are done.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

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