Now you're past my analysis level; are systems in which Cauchy sequences don't converge to a limit necessarily not Archimedean?
Regardless, in the sentence "{x_k} is a Cauchy sequence representing a real number x", how many predicates would you use? It's sounding like you would use two.
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mi'o bazu klama ti tu zi'oOn Thu, Sep 16, 2010 at 2:55 AM, Ross Ogilvie <oges007@gmail.com> wrote:
--But Cauchy sequences don't necessarily converge to a limit. I would say that convergence of a sequence to a limit should be a separate predicate, but my vocab isn't up to finding one. You could then indicate the limit of a Cauchy sequence with a tagged place.
RossOn Thu, Sep 16, 2010 at 3:18 PM, Ian Johnson <blindbravado@gmail.com> wrote:
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.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 September 2010 20:01:06 Ian Johnson wrote:"porsrkoci" and "pornkoci" are both good, and are different forms of the same
> I found myself being lazy in my analysis class having to repeatedly write:
> 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 clunky
> 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 thought
> of {porsrkoci}, which seems a bit better. Does anyone have any better
> ideas? Maybe something that isn't a fu'ivla?
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
longish, and "kocis.zei.porsi", which is also longish. I'd go
with "pornkoci".
I think the place structure should be "x1 (sequence) is a Cauchy sequence in
> 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 infinitum" in
> lojban.)
x1 (metric space)". I know a sequence of rational numbers which converges 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 number,
and there are sequences of rational numbers which are Cauchy sequences in one
metric but not in another.
Pierre
--
li fi'u vu'u fi'u fi'u du li pa
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