Register to reply

Cauchy sequences

by garyljc
Tags: cauchy, sequences
Share this thread:
garyljc
#1
Nov5-08, 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 ?
Phys.Org News Partner Mathematics news on Phys.org
'Moral victories' might spare you from losing again
Fair cake cutting gets its own algorithm
Effort to model Facebook yields key to famous math problem (and a prize)
Office_Shredder
#2
Nov5-08, 08:43 AM
Emeritus
Sci Advisor
PF Gold
P: 4,500
a(m) is the same sequence as a(n)
lurflurf
#3
Nov5-08, 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

HallsofIvy
#4
Nov5-08, 09:49 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,318
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.
garyljc
#5
Nov5-08, 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 ?
mathman
#6
Nov5-08, 02:58 PM
Sci Advisor
P: 6,031
Quote Quote by garyljc View Post
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 ?
It is easier to explain by example. Consider the sequence 1, 1/2, 1/3, 1/4,...

The sup is 1, while the lim sup is 0.
adnan jahan
#7
Nov10-08, 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
Pere Callahan
#8
Nov11-08, 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]


Register to reply

Related Discussions
Cauchy sequences Calculus 10
Cauchy Sequences Calculus & Beyond Homework 1
Show equivalence of two Cauchy sequences Introductory Physics Homework 0
Cauchy sequences in an inner product space Introductory Physics Homework 21
Cauchy sequences Introductory Physics Homework 2