How can I complete my proof: Sobolev space W^(1,p) is complete? Using Convergence theorem

Question

I'm trying to prove that W^(1,k) (R) is complete.

The steps i Had so far:

1. let {fn} be a cauchy sequence in W^(1,k). therefore {fn} and {dfn} are cauchy sequences in L^p(R), and therefore converge to f and g respectively.

2. I'm trying to show that g= df (reffering to weak deriviative).

for that, It would suffice to show that: integral (f* dpsi/dx) = lim integral (fn* d psi/dx).

I believe that for that kind of argument I'd need to use some convergence theorem (probably the donimated one?), but i failed to do so. Could you guide me through that?

Another attempt which failed was to use Holder's inequallity. I've reached: Integral ((f-fn)*d psi/dx) <= Integral |((f-fn)*d psi/dx)| <= ||f-fn||_p *||dpsi/dx||_q ->0 as n->inf So the conclusion was that the limit is <= 0. If i had lim = 0 I'd finish.

Thanks so much :)

Answer 1

You were done. Consider just $$\left| \int (f - f_n) \psi' dx \right| \leq \int |f-f_n||\psi'| dx \leq ||\psi'||_q||f-f_n||_p $$

So the limit is $0$