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Cauchy sequences 
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#1
Nov508, 08:20 AM

P: 106

By definition, a sequence a(n) has the Cauchy sequence if for eery E>0 ,there exist a natural number N such that Abs(a(n)  a(m) ) < E for all n, m > N
Could anyone tell me what is a(m) ? is it a subsequence of a(n) , or could it be any other non related sequence ? 


#2
Nov508, 08:43 AM

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PF Gold
P: 4,500

a(m) is the same sequence as a(n)



#3
Nov508, 08:47 AM

HW Helper
P: 2,263

a(m) and a(n) are not sequences they are elements of a sequence
Pehaps the difficulty will be eased by restating the definition differently a sequence is Cauchy if for any E>0 there exist a natural number N such that the difference between any two terms beyond N cannot exceed N or a sequence is Cauchy if for any E>0 there exist a natural number N such that Abs(a(N+n)  a(N+m) ) < E for all n,m that are natural numbers 


#4
Nov508, 09:49 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,300

Cauchy sequences
Neither a_m nor a_n in that is a sequence. They are, rather, any two numbers from the original sequence {a_i}, with, of course, m and n larger than N.



#5
Nov508, 11:53 AM

P: 106

OK thanks
One more question what's the difference between Lim sup a(n) and sup A(n) does the limit tells me something else ? 


#6
Nov508, 02:58 PM

Sci Advisor
P: 6,027

The sup is 1, while the lim sup is 0. 


#7
Nov1008, 10:52 PM

P: 95

what basically is the change that lim produced in sup
why it changes sup=1 to lim sup=0 and do this thing hold in all cases that lim sup is not the part of the sequence 


#8
Nov1108, 07:29 AM

P: 588

The limit superior of a sequence [itex](a_n)_{n\geq 0}[/itex] is the largest accumulation (or cluster) point of this sequence. An accumulation point is a number c such that in any neighbourhood of c there are infintely many members of the sequence. Analogously, the limit inferior is the least such accumulation point.
If [itex](a_n)_{n\geq 0}[/itex] is convergent, say with limit a, then [tex] \lim_{n\to\infty} {a_n} = \limsup_{n\to\infty}{a_n} = \liminf_{n\to\infty}{a_n} = a [/tex] 


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