You need to show that every Cauchy sequence converges in this norm. So let be Cauchy. Then as . But if the sum of two positive quantities tends to 0 then so do both of them individually. Therefore is Cauchy in , and is Cauchy in (with the sup norm). Since both those spaces are complete, there exist a real number c and a continuous function g such that and .

Now define and show that in the given norm.