Dini's theorem

In the mathematical field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is uniform. The theorem is named after Ulisse Dini.

Formal statement

If is a compact topological space, and is a monotonically increasing sequence of continuous real-valued functions on which converges pointwise to a continuous function, then the convergence is uniform. The same conclusion holds if is monotonically decreasing instead of increasing.
This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by the monotonicity. The limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous. The continuity of the limit function cannot be inferred from the other hypothesis

Proof

Let be given. For each, let, and let be the set of those such that. Each is continuous, and so each is open. Since is monotonically increasing, is monotonically decreasing, it follows that the sequence is ascending. Since converges pointwise to, it follows that the collection is an open cover of. By compactness, there is a finite subcover, and since are ascending the largest of these is a cover too. Thus we obtain that there is some positive integer such that . That is, if and is a point in, then, as desired.