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From: William Elliot on 16 Apr 2010 06:15
On Fri, 16 Apr 2010, David C. Ullrich wrote:
> <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...)
>
By limit point, do you mean a cluster point?

> What did the problem actually ask?
>
What are the actual definitions being used?
From: Eric on 16 Apr 2010 07:37
On 4月16日, 下午6時15分, William Elliot <ma...(a)rdrop.remove.com> wrote:
> On Fri, 16 Apr 2010, David C. Ullrich wrote:
> > <eric955...(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...)
>
> By limit point, do you mean a cluster point?
>
> > What did the problem actually ask?
>
> What are the actual definitions being used?

Tank you for your interest.My teacher talk about the "Conergence of
complex sequences"
Def.{Z_n} is convergent to the limit a iff
for all E>0, there exist N>0:for all n>N =>|Z_n-a|<E
this is almost mean that Z_n -> a as n -> infinite
and he introduce (Cauchy criterion) and use it to prove that
definition (Def)

and then he gave us this problem (How to prove that has one limit
point (Proof by contradiction ).

thank you,

Eric


From: Robert Israel on 16 Apr 2010 13:37
Eric <eric955308(a)yahoo.com.tw> writes:

> On 4=E6=9C=8816=E6=97=A5, =E4=B8=8B=E5=8D=886=E6=99=8215=E5=88=86, William
> =
> Elliot <ma...(a)rdrop.remove.com> wrote:
> > On Fri, 16 Apr 2010, David C. Ullrich wrote:
> > > <eric955...(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...)
> >
> > By limit point, do you mean a cluster point?
> >
> > > What did the problem actually ask?
> >
> > What are the actual definitions being used?
>
> Tank you for your interest.My teacher talk about the "Conergence of
> complex sequences"
> Def.{Z_n} is convergent to the limit a iff
> for all E>0, there exist N>0:for all n>N =3D>|Z_n-a|<E
> this is almost mean that Z_n -> a as n -> infinite
> and he introduce (Cauchy criterion) and use it to prove that
> definition (Def)
>
> and then he gave us this problem (How to prove that has one limit
> point (Proof by contradiction ).

I think there is a language barrier here. "Limit point" is not the same as
"limit". The actual problem must have been to prove that a sequence has at
most one limit.

OK, let's get you started with the proof by contradiction.

Hint: how can |Z_n - a| < E and |Z_n - b| < E produce a contradiction?
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
From: Eric on 16 Apr 2010 13:53
On 4¤ë17¤é, ¤W¤È1®É37¤À, Robert Israel
<isr...(a)math.MyUniversitysInitials.ca> wrote:
> Eric <eric955...(a)yahoo.com.tw> writes:
> > On 4=E6=9C=8816=E6=97=A5, =E4=B8=8B=E5=8D=886=E6=99=8215=E5=88=86, William
> > =
> > Elliot <ma...(a)rdrop.remove.com> wrote:
> > > On Fri, 16 Apr 2010, David C. Ullrich wrote:
> > > > <eric955...(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...)
>
> > > By limit point, do you mean a cluster point?
>
> > > > What did the problem actually ask?
>
> > > What are the actual definitions being used?
>
> > Tank you for your interest.My teacher talk about the "Conergence of
> > complex sequences"
> > Def.{Z_n} is convergent to the limit a iff
> > for all E>0, there exist N>0:for all n>N =3D>|Z_n-a|<E
> > this is almost mean that Z_n -> a as n -> infinite
> > and he introduce (Cauchy criterion) and use it to prove that
> > definition (Def)
>
> > and then he gave us this problem (How to prove that has one limit
> > point (Proof by contradiction ).
>
> I think there is a language barrier here. "Limit point" is not the same as
> "limit". The actual problem must have been to prove that a sequence has at
> most one limit.
>
> OK, let's get you started with the proof by contradiction.
>
> Hint: how can |Z_n - a| < E and |Z_n - b| < E produce a contradiction?
> --
> Robert Israel isr...(a)math.MyUniversitysInitials.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada- ÁôÂóQ¤Þ¥Î¤å¦r -
>
> - Åã¥Ü³Q¤Þ¥Î¤å¦r -

Thank you very much.
Maybe we should suppose it has two limit point.And find the
contradiction.Sorry I have no idea.
From: Dave L. Renfro on 16 Apr 2010 14:19
Robert Israel wrote:

>> Hint: how can |Z_n - a| < E and |Z_n - b| < E produce a contradiction?

Eric wrote:

> Thank you very much.
> Maybe we should suppose it has two limit point.And find the
> contradiction.Sorry I have no idea.

|Z_n - a| < E is the interior of the circle with center a
and radius sqrt(E). |Z_n - b| < E is the interior of the
circle with center b and radius sqrt(E). Draw a diagram
of the two circular regions when E is very small . . .

Dave L. Renfro
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