## Dini’s theorem and related problems

Let be two real numbers and a sequence of continuous functions which converge pointwise to a continuous function .

1. **Dini’s Theorem.** Suppose that the sequence has the property that . Prove that the convergence is uniform.

2. Suppose that every function is increasing. Prove that the convergence is uniform.

3. If every function is convex on , prove that the convergence is uniform on every compact interval in .

Proofs: 1. Denote and . Notice that since is increasing must be decreasing, so . Then each is open and since converges pointwise to zero it follows that is a cover of . Since is compact, we can extract a finite subcover. The sets in this subcover are ordered by inclusion, so the greatest of them is the whole . Therefore there exists such that for we have . This implies the fact that the convergence is uniform.

2. The limit function is obviously increasing. Pick . We can find such that (if we go outside the interval, extend by making them constant outside , keeping continuity. From the convergence of to we deduce the existence of such that for every we have

Now if then and . Therefore can move in an interval of length containing which means .

Obviously the sets cover . Pick a finite subcover and the maximum of for in that subcover and we have that for every . Thus the convergence is uniform.

3. Let . If are convex then is also convex. As a consequence, the functions

are well defined and increasing for small enough. Furthermore the pointwise convergence implies the pointwise convergence . Apply the second part to conclude that the convergence is uniform. This implies that the convergence is also uniform.