| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
ok, lets see who'll get this one:
when is a Cauchy sequence not convergent ?
a sequence {X(n)} is called a cauchy sequence if it satisfies
the following:
for any e>0 there exist N=N(e) , (i.e. N may or may not depend on e)
and N>0 , s.t. || X(n)- X(m) || < e for all m,n >= N
it is interesting that such sequence is not always convergent !
i probably need to say more about what the elements of the sequence
belong to , i.e. more about the context of the problem ? do you
need this information?
note that every convergent sequence is a cauchy sequence, but not
otherway around.
can you define one such cauchy sequence that is not convergent ?
/Nasser
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 1506.1 | ALLVAX::JROTH | I know he moves along the piers | Sun Oct 20 1991 20:23 | 8 | |
This really hinges on whether the space your sequence comes from
is complete or not; if it is then yes - every Cauchy sequence
does converge to an element of the space.
You should be able to make an example with rational numbers since
that set isn't complete.
- Jim
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| 1506.2 | example | STAR::ABBASI | Sun Oct 20 1991 23:38 | 13 | |
That is right Jim.
---
one such example could be sequence that converges to \/2
in metric space R, looking at (1.4,1.41,1.414,....) with successive
approximation to \/2 , this is not convergent in Q however, since \/2
is not in Q. it is however convergent in R, since the limit L is in R.
another name to the complete normed space is Banach space.
/Nasser
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