Is there a space where no convergent Cauchy sequences exist?
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I understand every non-complete space can be made complete by including the limit points of convergence of Cauchy sequences. I assume that the opposite process is possible as well. But, do spaces where no convergent Cauchy sequence exists possible naturally without any such artificial construction?
cauchy-sequences
asked Dec 5 '18 at 2:49
Kiran GarimellaKiran Garimella
1
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add a comment |
$begingroup$
I understand every non-complete space can be made complete by including the limit points of convergence of Cauchy sequences. I assume that the opposite process is possible as well. But, do spaces where no convergent Cauchy sequence exists possible naturally without any such artificial construction?
cauchy-sequences
asked Dec 5 '18 at 2:49
Kiran GarimellaKiran Garimella
1
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$begingroup$
If $x$ is a point in the space, then $x,x,x, dots$ is a Cauchy sequence which converges to $x$.
$endgroup$
– platty
Dec 5 '18 at 2:51
add a comment |
$begingroup$
I understand every non-complete space can be made complete by including the limit points of convergence of Cauchy sequences. I assume that the opposite process is possible as well. But, do spaces where no convergent Cauchy sequence exists possible naturally without any such artificial construction?
cauchy-sequences
asked Dec 5 '18 at 2:49
Kiran GarimellaKiran Garimella
1
$endgroup$
I understand every non-complete space can be made complete by including the limit points of convergence of Cauchy sequences. I assume that the opposite process is possible as well. But, do spaces where no convergent Cauchy sequence exists possible naturally without any such artificial construction?
cauchy-sequences
cauchy-sequences
asked Dec 5 '18 at 2:49
Kiran GarimellaKiran Garimella
1
asked Dec 5 '18 at 2:49
Kiran GarimellaKiran Garimella
1
asked Dec 5 '18 at 2:49
Kiran GarimellaKiran Garimella
1
asked Dec 5 '18 at 2:49
Kiran GarimellaKiran Garimella
1
asked Dec 5 '18 at 2:49
Kiran GarimellaKiran Garimella
1
1
$begingroup$
If $x$ is a point in the space, then $x,x,x, dots$ is a Cauchy sequence which converges to $x$.
$endgroup$
– platty
Dec 5 '18 at 2:51
add a comment |
$begingroup$
If $x$ is a point in the space, then $x,x,x, dots$ is a Cauchy sequence which converges to $x$.
$endgroup$
– platty
Dec 5 '18 at 2:51
$begingroup$
If $x$ is a point in the space, then $x,x,x, dots$ is a Cauchy sequence which converges to $x$.
$endgroup$
– platty
Dec 5 '18 at 2:51
$begingroup$
If $x$ is a point in the space, then $x,x,x, dots$ is a Cauchy sequence which converges to $x$.
$endgroup$
– platty
Dec 5 '18 at 2:51
add a comment |
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$begingroup$
If $x$ is a point in the space, then $x,x,x, dots$ is a Cauchy sequence which converges to $x$.
$endgroup$
– platty
Dec 5 '18 at 2:51