Completion

NONEXAMINABLE in Part II
For any normed space there is a completion of
meaning a Complete Normed Space in which is dense

Proof

Let if as
Let be the set of all Cauchy Sequences in
quotiented by the equivalence relation
This is naturally a Vector Space
It is also a Normed Space with norm

Suppose is some sequence in
This converges to the diagonal
Thus is Complete

Also is dense in via