Existence of Cauchy and convergent sequences

Question

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

Answer 1

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.