For a Cauchy sequence that does not converge, take a look at , where

For the completion, you should expect it to be a space in which the formula for a point to belong to the space coincides with the formula for the norm. So I would expect the completion to be the space

Edit.No, I was wrong about the completion of the space. I think it should be That is complete in the given norm, and it contains as a dense subspace.