On Thu, Sep 16, 2010 at 6:49 PM, Ross Ogilvie <= ;oges007@gmail.com> wrot= e:

Given xorxes' word for convergence, I'd translate "{x_k} is a = Cauchy sequence representing a real number x" as
{x_k} jbize'a x

Because in your context of real analysis all con= vergent sequences are Cauchy sequences (and vice versa). In a more general = metric space context

"{x_k} is a Cauchy sequence in metric spa= ce Y with limit x"
{x_k} pornkoci Y fi'o jbize'a x

And to answer your maths que= stion. Consider the space of rational numbers. Then there are sequences of = rational numbers that converge to irrational numbers. So considered in Q al= one, they are Cauchy sequences (because all the terms get closer and closer= together) but they fail to have a limit (they converge to a point outside = the set). However the rationals are archimedean. This is basically because = Cauchy sequences are powerful enough to find 'holes' in the space (= for this example, you can find holes in the rationals where you would norma= lly have the irrationals)

Ross

On Fri, Sep 17, 2010 at 3:31 AM, Ian Johnson <= ;blindbravado@g= mail.com> wrote:

<= /div>
Now you're past my analysis level; are systems in which Cauchy sequence= s don't converge to a limit necessarily not Archimedean?

Regardl= ess, in the sentence "{x_k} is a Cauchy sequence representing a real n= umber x", how many predicates would you use? It's sounding like yo= u would use two.

--

mi'o bazu klama ti tu zi'o

<= div class=3D"gmail_quote">
On Thu, Sep 16, 2010 at 2:55 AM, Ross Ogilvi= e <oges007@gmail.com> wrote:

<= div>

But Cauchy sequences don't necessarily converge to a= limit. I would say that convergence of a sequence to a limit should be a s= eparate predicate, but my vocab isn't up to finding one. You could then= indicate the limit of a Cauchy sequence with a tagged place.

Ross

On T= hu, Sep 16, 2010 at 3:18 PM, Ian Johnson <blindbravado@gmail.com&= gt; wrote:

<= /div>

That makes sense, but should that really be the x2? Or should there be= another predicate that we use to relate numbers and sequences? For example= , we could say:
x1 is a Cauchy sequence converging to limit x2 in= metric space x3.=A0

mi'o bazu klama ti tu zi'o
=

On Wed, Sep 15, 2010 at 10:48 PM, Pierre Abbat <phma@= phma.optus.nu> wrote:

On Wednesday 15 Se= ptember 2010 20:01:06 Ian Johnson wrote:
> I found myself being lazy in my analysis class having to repeatedly wr= ite:
> Let x in R. Suppose {x_k} is a Cauchy sequence representing x.
> I was trying to come up with a good word to use to represent this clun= ky
> relation, that is:
> x1 is a Cauchy sequence representing the real number x2.
> The thing I came up with first was pretty bad, but I didn't have a=
> dictionary on me. It was {listrkoci}. Once I got to a dictionary I tho= ught
> of {porsrkoci}, which seems a bit better. Does anyone have any better<= br> > ideas? Maybe something that isn't a fu'ivla?

"porsrkoci" and "pornkoci" are both good, and are= different forms of the same
word (though the Book doesn't say that different rafsi of one gismu are=
equivalent, except in lujvo). The alternatives are a lujvo, which would be<= br> longish, and "kocis.zei.porsi", which is also longish. I'd go=
with "pornkoci".

> To clarify, this should hold, if broda is assigned to this relation: > li pa ce'o li pa fi'u re ce'o li pa fi'u ci ce'o .= .. broda li no
> (sorry that I don't know a good way to say "et cetera ad infi= nitum" in
> lojban.)

I think the place structure should be "x1 (sequence) is a Cauchy= sequence in
x1 (metric space)". I know a sequence of rational numbers which conver= ges to
+3 in the real numbers and to -3 in the 2-adic numbers. There are Cauchy sequences of rational numbers which don't converge to any rational numb= er,
and there are sequences of rational numbers which are Cauchy sequences in o= ne
metric but not in another.

Pierre
--
li fi'u vu'u fi'u fi'u du li pa

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