What's the black square for Jane?

Yeah, that’s what it means.

Theorem 1 says that two Cauchy sequences can be added term by term and the result is another Cauchy sequence. In view of this, it makes sense to define addition on

by

   

Last edited by JaneFairfax (2009-02-06 12:03:24)

I suppose multiplicative inverses come next, that's a fun proof.

In fact, as we shall see, it’s more than just a field. But I’m 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.

Last edited by JaneFairfax (2009-02-09 13:46:42)

The hat notation is used in Sutherland’s Introduction to Metric and Topological Spaces. I use it myself because I think it’s cute.

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.

I AM VERY SORRY, FOLKS. I JUST NOTICED A GAPING HOLE IN MY PROOF OF THEOREM 6 WHICH I MUST PLUG RIGHT AWAY.

JaneFairfax wrote:

I NEED TO CONFIRM THAT THIS SEQUENCE IS CAUCHY (which I didn’t 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.

Last edited by JaneFairfax (2010-12-12 01:25:08)

Great manuveur