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.

Formal statement

If X is a compact topological space, and is a monotonically increasing sequence of continuous real-valued functions on X which converges pointwise to a continuous function f, then the convergence is uniform. The same conclusion holds if is monotonically decreasing instead of increasing. The theorem is named after Ulisse Dini.
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.

Proof

Let ε > 0 be given. For each n, let gn = f − fn, and let En be the set of those x ∈ X such that gn < ε. Each gn is continuous, and so each En is open. Since is monotonically increasing, is monotonically decreasing, it follows that the sequence En is ascending. Since fn converges pointwise to f, it follows that the collection is an open cover of X. By compactness, there is a finite subcover, and since En are ascending the largest of these is a cover too. Thus we obtain that there is some positive integer N such that EN = X. That is, if n > N and x is a point in X, then |f − fn| < ε, as desired.