idea for the completion of a metric space While doing the proof of the existence of completion of a metric space, usually books give an idea that the missing limit points are added into the space for obtaining the completion. But I do not understand from the proof where we are using this idea as we just make equivalence classes of asymptotic Cauchy sequences and accordingly define the metric.

For a metric space $\langle T, d\rangle$ to be complete, all Cauchy sequences must have a limit. So we add that limit by defining it to be an "abstract" object, which is defined by "any Cauchy sequence converging to it".
We have two cases:


*

*The Cauchy sequence already had a limit in $T$. In this case there is no need to add new points, and we identify that abstract object to the already existing point.

*The Cauchy sequence did not converge in $T$. Then you add this "object" to your space, and define distance accordingly. You can prove using triangle equality that you can choose any "equivalent" Cauchy sequence, and the metric will be the same.
The important point is that the point we add are not real points, they are just abstract objects, which have some property that make them behave well under your tools and language.
