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From: Eric on 15 Apr 2010 23:02
How to prove the sequence just has one limit point (Proof by
contradiction ).
From: Chip Eastham on 15 Apr 2010 23:12
On Apr 15, 11:02 pm, Eric <eric955...(a)yahoo.com.tw> wrote:
> How to prove the sequence just has one limit point (Proof by
> contradiction ).

Hi, Eric:

Did you have a particular sequence in mind?

Or a particular property like "Cauchy sequence"?

regards, chip
From: Eric on 15 Apr 2010 23:17
On 4月16日, 上午11時12分, Chip Eastham <hardm...(a)gmail.com> wrote:
> On Apr 15, 11:02 pm, Eric <eric955...(a)yahoo.com.tw> wrote:
>
> > How to prove the sequence just has one limit point (Proof by
> > contradiction ).
>
> Hi, Eric:
>
> Did you have a particular sequence in mind?
>
> Or a particular property like "Cauchy sequence"?
>
> regards, chip

Yes,it's Cauchy sequence.Thank you
From: William Elliot on 16 Apr 2010 03:30
On Thu, 15 Apr 2010, Eric wrote:

> How to prove the sequence just has one limit point (Proof by
> contradiction ).
>
That a sequence can converge to at most
one point is a property of Hausdorff spaces.

Yes. Assume a sequence s converges to x and to y and x /= y.
Then there's some disjoint open sets U,V separating x and y.

As s must eventually be in U and also eventually in V,
a friendly contradiction arises forth for the finish.










From: David C. Ullrich on 16 Apr 2010 06:02
On Thu, 15 Apr 2010 20:02:24 -0700 (PDT), Eric
<eric955308(a)yahoo.com.tw> wrote:

>How to prove the sequence just has one limit point (Proof by
>contradiction ).

_What_ sequence? Some sequences have more than one
limit point.

A given sequence has at most one limit (some sequences do not
have a limit, because they have more than one limit point...)

What did the problem actually ask?


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