Metric Spaces, The Completion of a Metric Space

The Completion of a Metric Space

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.