Re: notation: sup, inf, limit, sequences

In article <150720082137136894%plsperry@[EMAIL PROTECTED]
>,
Paul Sperry wrote:

> In article <slrng7jspg.6um.dbast0s@[EMAIL PROTECTED]
>, Daniel Chicayban
> Bastos <dbast0s@[EMAIL PROTECTED]
> wrote:
>
>> In article <150720081458151297%plsperry@[EMAIL PROTECTED]
>,
>> Paul Sperry wrote:
>> 
>> > In article <slrng7pg7f.leo.dbast0s@[EMAIL PROTECTED]
>, Daniel
>> > Chicayban Bastos <dbast0s@[EMAIL PROTECTED]
> wrote:
>> >
>> >> In article <140720081719437654%plsperry@[EMAIL PROTECTED]
>,
>> >> Paul Sperry wrote:
>> >> 
>> >> > In article <slrng7mg9o.cu0.dbast0s@[EMAIL PROTECTED]
>, Daniel
>> >> > Chicayban Bastos <dbast0s@[EMAIL PROTECTED]
> wrote:
>> >> >
>> >> >> In article <130720081727235868%plsperry@[EMAIL PROTECTED]
>,
>> >> >> Paul Sperry wrote:
>> >> >> 
>> >> >> > In article <slrng7i5ve.5gn.dbast0s@[EMAIL PROTECTED]
>, Daniel
Chicayban
>> >> >> > Bastos <dbast0s@[EMAIL PROTECTED]
> wrote:
>> >> >> >
>> >> >> >> In article <130720081359512555%plsperry@[EMAIL PROTECTED]
>,
>> >> >> >> Paul Sperry wrote:
>> >> >> >> 
>> >> >> >> > In article <slrng7hc69.5gn.dbast0s@[EMAIL PROTECTED]
>, Daniel
Chicayban
>> >> >> >> > Bastos <dbast0s@[EMAIL PROTECTED]
> wrote:
>
> [...]
>
>> > Well, let's take a look at the usual "Calculus" definition of
>> > continuity, in the extended Reals, of a function f at oo. It would
say
>> > lim(f(x) : x -> oo) = L and f(oo) = L. Hmmm, well, OK. What does
_that_
>> > say? For every e > 0 there is a d > 0 such that if 0 < |x - oo .....
>> > Oh, oh. What's x - oo? Turns out that any attempt to define
arithmetic
>> > in the extended Reals really messes up the algebra. That's why the
>> > extended Reals, although much beloved by analysts and topologists, is
>> > much despised by algebraists.
>> >
>> > (There _is_ a metric for the extended Reals but it is _not_ an
>> > extension of the usual metric on the Reals --- I think.)
>> 
>> I think I see the problem. If the metric on the extended reals is not
an
>> extension of the usual metric, then we have a distinguished system,
yes?
>> I mean, what's valid in one is not necessarily valid on the other.
>
> Correct. A sequence may converge according to one metric but not
> converge according to the other. See Brian's most recent response to me
> in this thread.

Yes, I begin to notice that it all depends on the topology. I remember
seeing a non-Hausdorff space in which a sequence would converge to any
point in its neighborhood of convergence. It was trivial example. If I
recall correctly about this, then the space X would have a topology ({},
X). So if s(n) converges in X, then it must converge to any point in X,
after all there aren't any other choices. True?

>> It gets tricky. I say that when we're working with a sequence and it
>> converges to a real L, fine, we're in the usual real number system even
>> though we're using the undefined +oo while keeping the idea of limit
>> precise --- we've talked about this once ---, but if the sequence
>> converges to +oo (thus converging in the extended reals), we have a
>> sequence that really belongs in another system, yes?
>> 
>> Seems divisive. My preferred way to think of mathematics is always as
an
>> extension of earlier things; as much as possible. The usual order on
the
>> reals doesn't extend to the complex plane, but that's fine as we have
>> good reasons for it and an extension need not imply that we bring all
>> properties --- on the contrary; if we're getting deeper, it makes sense
>> that we lose particular properties keeping the more general ones.
>
> It is not as bad as it seems - you are trying to "extend" the wrong
> thing.. As far as I know, no one is particularly interested in a metric
> on R* (= the extended reals).
>
> However, all is not lost - far from it. We endow R* with the order
> topology.
>
> I don't know how much you know about topology so this will be sketchy
> and vague. (While I'm spending your money, you can get the classic
> "General Topology" by John Kelley pretty inexpensively from Amazon - it
> is worth having if only for its set theory appendix.)
>
> Anyway, consider the left and right rays in R* : {x : x < a} and 
> {x : x > a} for any a in R* (of course if a = oo the right ray is
> empty, etc.). 
>
> Consider the set consisting of all rays and all open intervals, (a, b) 
> for a and b in the usual Reals. Call this collection B. We can say a 
> sequence a_n converges to L in R* if, whenever L belongs to an 
> element U of B, the sequence eventually lies in B - that is, there is 
> an N such that if n > N, a_n is in U.
>
> If you think about it that says that if L is an ordinary Real
> convergence means just what it always has but for example, 
> 1, 2, ..., n, ... really _does_ converge to +oo according to this
> definition.

Hm. Let's see. We allow that in the topology defined above, yes? You
said a can be +oo, so there is an interval such as (p, oo) in B which as
long as we find an N such that for all n > N, a_n is in (x, oo), then it
converges there. So, for 1, 2, ..., n, we find N = 2 and U = (1, oo)
which works. Does this make sense?

> For your amusement and edification look up "one point compactification"
> - Wikipedia will do. There, oo is used as _both_ ends of R!

Don't know intuitively what a compact space is, and that seems to be a
requirement for reading this article.