Yes. You can find in many books on topology this construction, Kuratowski, Topology vol.I p.410, for instance. If X is a metric space, you take a space Y of equivalence classes of Cauchy sequences in X with respect to an appropriate relation with a natural metric. Then X is embedded in Y as a dense subset: a class of constant sequence x is a corresponding element in Y.

Thanks for your answer but my question is if I can take other family of sequences, not only a family of Cauchy sequences, for having another way of completing the space. Is there a general method for that?

I'm afraid your question is not precisely defined. If you want to find another method for embedding a metric space into a complete one, then there are some but without using sequences.

Using sequences too but instead Cauchy sequences, another type of sequences.

The questions needs to be clarified. However, the most general technique is to use sets of functions from the space X which we want to complete into a natural complete object E, and endow this set with the product topology from E. The space X gets embedded into it via the evaluations, and then take the closure of this embedded subset which usually is the smallest complete object containing X. For several examples of this kind, please look into the book Hewitt Nachbin Spaces by Maurice Weir. He produces a fairly simple and readable account of Mrowka's work on E-completely regular spaces.

I think that a natural question would be this: if we use some other notion of convergent sequence than "Cauchy", do we get any more limit points in our completed space. I suspect that since Cesaro convergent sequences converge to real numbers, you are unlikely to obtain anything new.

A general situation is this: a metric space X is densely embedded into a space Y, which is assumed to be metrizable. Then it is first-countable at each point of Y\X whereas each boundary point is the limit of some net in X. A net that converges to a point y with a countable neighborhood base must contain a converging subsequence.

No matter how Y has been obtained, it can be obtained alternatively by considering suitable Cauchy sequences in X. Note that I do not even require Y to be completely metrizable.