I'm not really sure if I'm approaching this correctly, since I'm not using linearity. What exactly do I need to prove here?
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.