Noun
Cauchy sequence (plural Cauchy sequences)
(mathematical analysis) Any sequence
x
n
{\displaystyle x_{n}}
in a metric space with metric d such that for every
ϵ
>
0
{\displaystyle \epsilon >0}
there exists a natural number N such that for all
k
,
m
≥
N
{\displaystyle k,m\geq N}
,
d
(
x
k
,
x
m
)
<
ϵ
{\displaystyle d(x_{k},x_{m})<\epsilon }
.
A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. Source: Internet
A topological vector space where every Cauchy sequence converges is sequentially complete but may not be complete (in the sense Cauchy filters converge). Source: Internet
By definition, in a Hilbert space any Cauchy sequence converges to a limit. Source: Internet
Therefore, (x n ) is a Cauchy sequence in E, converging to some limit point a in E, because E is complete. Source: Internet
For instance, the set of rational numbers is not complete, because e.g. 2 is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it. Source: Internet
In the real numbers every Cauchy sequence converges to some limit. Source: Internet