I don't believe that for any linear subspace of a Banach space, every convergent sequence is Cauchy. Maybe I'm wrong?
Convergent sequences are always Cauchy. Cauchy sequences always converge in complete spaces (by definition, almost, depending on the author), but they don't have to converge if the space is not complete.
There isn't much to this proof, and I think you're certainly got the general idea. Closedness and completeness, as this proof shows, are pretty much the same thing.