We will use the following result proved in analysis. It is (a special case of) the Arzela-Ascoli Theorem.

Theorem A.4.1. Let ? C R ^s be a closed bounded set, and let u _m : ??C ⁿ denote a sequence of functions with the following properties:

1. For each ?>0, there is a ?>0 independent of m with

   
   

2. There is a K independent of m with

   
   

Then there is a continuous function u: ?? C ⁿ, and a sequence of indices m _j ?? with

To apply the result, assume that we have a sequence of functions

which is uniformly smooth; i.e., u _m ? C ^? for all m, and for all nonnegative integers p, q there is a constant C(p, q) independent of m with

We want to show:

Theorem A.4.2. There is a function u ? C ^?, and a sequence m _j ?? with

In short, a subsequence of u _m converges along with all its derivatives to a C ^? -limit.

Proof. By the Arzela-Ascoli Theorem there is a continuous u and a sequence m=m _j ?? with

Now apply the Arzela-Ascoli result to the sequence

and obtain the existence of a continuous ? and a sequence j _k ?? with

From

we conclude that for m= m _jk ??,

Hence u is differentiable w.r.t. x, and

Let us show...