Cauchy sequence
is Cauchy when : all terms eventually lie within of each other.
Examples
Example 1.18
-
. Assume WLOG , Then
-
is not a Cauchy sequence, because if , for any , then
The definition fails for .
-
on defined by truncation of decimal expansion of :
This is Cauchy, since for WLOG , we have
This sequence does not converge over , but it does converge over .
Cauchy sequences are bounded
Proof of boundedness
Take . There is an with for all , hence there. Since is a fixed number, the whole sequence is bounded by
Convergence implies Cauchy
Lemma 1.21
If , then is Cauchy.
Related
Stated in
- Definition 1.17 (Cauchy Sequence)§1.3 Cauchy Sequences
- Example 1.18§1.3 Cauchy Sequences
- Lemma 1.19§1.3 Cauchy Sequences
- Lemma 1.21§1.3 Cauchy Sequences