Cauchy Sequences: Definition & a(m) Clarification

[garyljc]

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 ?

 

[Office_Shredder]

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

 

[lurflurf]

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]

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]

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]

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]

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]

The limit superior of a sequence $(a_n)_{n\geq 0}$ 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 infinitely many members of the sequence. Analogously, the limit inferior is the least such accumulation point.

If $(a_n)_{n\geq 0}$ is convergent, say with limit a, then
$$\lim_{n\to\infty} {a_n} = \limsup_{n\to\infty}{a_n} = \liminf_{n\to\infty}{a_n} = a$$