The Completion of a Metric Space

C[E] := {a : (an)n∈N is a Cauchy sequence in (E, d)} . You should think of C[E] as a new space where each point a ∈ C[E] is a Cauchy sequence a = (an)n∈N from (E, d). Note that for each x ∈ E, we can define x ∈ C[E] by letting xn = x for all n ∈ N. This gives us a way to think of C[E] as containing our original space E, which we will make more precise in Section 3. For a, b ∈ C[E] we define D(a, b) := lim n→∞ d(an, bn), (you will show that this limit exists in Exercise 1.1). This might seem like an unnecessarily complicated thing to do, but there is a good intuitive reason for why to consider C[E] (see Section 5). The hope with this initial construction is that (C[E], D) is a complete metric space, but, as will be seen in part (v) of Exercise 1.2, D fails to even be a metric. Hence, we will have to make some adjustments to this initial construction, which we shall undertake in the following sections.