Definition: A metric space is said to be complete if every Cauchy sequence is convergent.
Now, my question is: Is there a complete metric space which has no Cauchy sequence?
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3Note that eventually constant sequences are Cauchy. – André Nicolas Jun 19 '16 at 20:35
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The countable set of distinct points you mean . Thanks for answering – Haitham Abu Sarries Jun 19 '16 at 20:40
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As others have pointed out, in any metric space every eventually constant sequence is Cauchy. However, in a discrete metric space these are the only Cauchy sequences. – Brian M. Scott Jun 19 '16 at 20:43
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A constant sequence is Cauchy and convergent. Any such metric space would have to be a null set.
ncmathsadist
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2No, a null set. If there is a point, make the constant sequence there. – ncmathsadist Aug 25 '21 at 19:49
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No, there is always a cauchy sequence in any (nonempty) metric space: take $a_n=x$ for some element $x$ of the space. This is trivially Cauchy.
jgon
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What about the eventually constant sequences , does it have Cauchy sequences ? – Haitham Abu Sarries Jun 19 '16 at 20:48
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2@HaithamAbuSarries An eventually constant sequence is Cauchy. Just look at the definition. After some point all the differences are $0$. – user4894 Jun 19 '16 at 21:26