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I understand the definition of what is a Cauchy sequence, and what is a convergent sequence,so my question is in many topics involving functional analysis and/or Sobolev spaces, it is assumed there exists a Cauchy sequence, or a convergent sequence, what is the justification for the same? In which spaces can we assume existence of a Cauchy sequence, or even a convergent sequence. Thanks, Sandy

sandy kuks
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1 Answers1

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You need a notion of distance, to be able to speak of Cauchy sequences. So, you must be in at least a metric space. Take a point $x$ in your metric space. Then, the constant sequence $x_n=x$ for all $n$, is both Cauchy and convergent.

So, one could say that whenever it makes sense to speak about Cauchy sequences, one exists.

Lilla
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  • Thanks @warm_wish, in metric spaces (assuming it is also given to be a linear space), perhaps we can then talk about existence of Cauchy sequences with distinct elements of the sequence? – sandy kuks Oct 03 '21 at 17:25
  • What field comes with the vector space structure? – Lilla Oct 04 '21 at 05:13