JaneFairfax wrote:

I NEED TO CONFIRM THAT THIS SEQUENCE IS CAUCHY (which I didnt do)!!

is essentially a sequence of the form . In view of Theorem 3, one just needs to prove that is Cauchy. Note that this is not true for all Cauchy sequences only for non-null sequences . And to prove that the sequence of reciprocal terms is Cauchy, I shall need to use the result of Theorem 7 which means that I ought to have presented Theorem 7 before Theorem 6.Watch this space.

]]>This is an important result. It means that we can unambiguously define an order relation in

by

The corollary to Theorem 8 says that this order relation is well defined.

]]>Sutherland also writes for the sequence . However I choose not to drop the peripheral adjuncts, and write as a reminder that I am talking of the whole Cauchy sequence, not just the th term. ]]>

We now define

to be the set of all non-null Cauchy sequences in satisfying property in the theorem above. Certainly is nonempty since . is thought of as the set of all positive Cauchy sequences of rational numbers. ]]>In fact, as we shall see, its more than just a field. But Im going to build up the pieces slowly.

http://www.mathisfunforum.com/viewtopic.php?id=10480

Did you remember that thread? Never forget anything I post you never know when it may prove useful one day.

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http://z8.invisionfree.com/DYK/index.php?showtopic=192 ]]>

So term-by-term multiplication of Cauchy sequences also gives rise to Cauchy sequences. Hence multiplication in can also be validly defined.

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For this, I will need to assume the result (which I shall prove in a moment) that all Cauchy sequences are bounded.

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