Metric Spaces, The Completion of a Metric Space

The Completion of a Metric Space

Given a metric space S, let T be the set of Cauchy sequences taken from S. Embed S in T using constant sequences, just as we carried the rationals into the reals.

Let a and b be two Cauchy sequences taken from T. Let d(a,b) be the sequence d_j = |a_j,b_j|. Given ε, choose n so that both tails are within ε of a_n and b_n respectively. Now d_j is within 2ε of d_n. In other words, d is a Cauchy sequence of real numbers. Since the reals are complete, this is a real number. Let this be the distance between a and b.

Verify that the triangular inequality holds; it is inherited from S. And the distance from a to b is the distance from b to a. Thus we have a valid pseudo metric. Turn T into a proper metric space by clumping equivalent sequences together. In other words, alll sequences that are distance 0 apart become one point. We did the same thing for the reals. The result is a metric space, the completion of S.

Is T a complete metric space? It is, using the same proof as the reals. The completion of any metric space is complete.