Dini's theorem

In the mathematical field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges on a compact space, it converges uniformly.

Formal statement

If X is a compact topological space, and { fn } is a monotonically increasing sequence (meaning fn(x) ≤ fn+1(x) for all n and x) 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 { fn } 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. Note also that 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( x ) < ε. Each gn is continuous, and so each En is open (because each En is the preimage of gn, a continuous function). Since { fn } is monotonically increasing, { gn } is monotonically decreasing, it follows that the sequence En is ascending. Since fn converges pointwise to f, it follows that the collection { En } is an open cover of X. By compactness, 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( x ) − fn( x )| < ε, as desired.

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