The Cauchy criterion states that:
A sequence is Cauchy if and only if it converges.
Almost all proofs I can find for this theorem use the fact that:
A Cauchy sequence is bounded.
However, both the definitions of Cauchy sequence and converged sequence only use the metric of the system, which gives the distance between elements. While bounded property measures the elements themselves, use the norm of the system.
It is true that we may define consistent metric and norm. However, it is not necessary. Thus why we need the 'extra' norm to prove a statement only involves metric?
The question is: if a sequence (either Cauchy or Converged) comes from a set that only defining metric (and the metric may not satisfy translation invariance, thus it can not induce a norm, so that we can not use it to give the bounded property of a Cauchy sequence), can we still prove the Cauchy criterion?