w**a
发帖数: 1024 | 1 a cauchy sequence defined in matric space (X,d)
we know any cauchy sequence is bounded.
But this sequence also has infinite number of elements?
infinite bounded sequence can have NO accumulation point? Thanks. |
r*****r
发帖数: 630 | 2 It depends on if the metric space is compelete
【在 w**a 的大作中提到】
: a cauchy sequence defined in matric space (X,d)
: we know any cauchy sequence is bounded.
: But this sequence also has infinite number of elements?
: infinite bounded sequence can have NO accumulation point? Thanks.
|
x*****d
发帖数: 427 | 3 If the metric space is the set Q of rational numbers,
you see the Cauchy sequence 3.1, 3.14, 3.141, ... which
does not converge in the space.
【在 w**a 的大作中提到】
: a cauchy sequence defined in matric space (X,d)
: we know any cauchy sequence is bounded.
: But this sequence also has infinite number of elements?
: infinite bounded sequence can have NO accumulation point? Thanks.
|
w**a
发帖数: 1024 | 4 assume the space is complete.
Must any Cauchy sequence converge to some limit? seems that's the definition
for completeness?
【在 x*****d 的大作中提到】
: If the metric space is the set Q of rational numbers,
: you see the Cauchy sequence 3.1, 3.14, 3.141, ... which
: does not converge in the space.
|
z***u
发帖数: 1 | 5
Hilbert Space is such a space.
【在 w**a 的大作中提到】
: assume the space is complete.
: Must any Cauchy sequence converge to some limit? seems that's the definition
: for completeness?
|
