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.
 A: One way to construct the completion of a metric space $(X,d)$ is by isometrically embedding it into a large metric space, that is known to be complete already. 
A classical way is due to (IIRC) Banach: define $CB(X)$ to be the set of all bounded continuous real-valued functions on $X$ with metric from the supremum norm. This is a Banach space (so a complete metric space in particular). 
Fix $p \in X$; we can define for every $x \in X$ the function $f_x: X \rightarrow \mathbb{R}$ by $f_x(y) = d(x,y) - d(p,y)$. By standard facts this is a continuous function on $X$ and it is bounded by $d(x,p)$ (from the triangle inequality). It's not too hard to verify that $F(x) = f_x$ defines an isometry of $X$ into $CB(X)$. 
The closure of $F[X]$ in $CB(X)$ is then a completion of $X$: it contains $X$ ( or really $F[X]$) as an isometric dense subset and is itself complete in its inherited metric as a closed subset of a complete metric space.
The extra points we add to $X$ to make it complete are then the points in $CB(X) \setminus F[X]$, as it were. 
It's not as elegant as the usual construction (equivalence classes of Cauchy sequences) because we need to have $\mathbb{R}$ constructed first and know it to be complete in its usual metric (to get completeness of $CB(X)$). The equivalence class approach is more self-contained, though a bit more abstract.  
A: 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.
