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 (meaning for all and x\in X) of continuous real-valued functions on X which converges pointwise to a continuous function, 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. The continuity of the limit function cannot be inferred from the other hypothesis (consider x^n in [0,1].)

Proof

Let be given. For each, let g_n=f-f_n, and let E_n be the set of those x\in X such that. Each g_n is continuous, and so each E_n is open (because each E_n is the preimage of the open set under g_n, a continuous function). Since is monotonically increasing, is monotonically decreasing, it follows that the sequence E_n is ascending (i.e. for all ). Since converges pointwise to f, it follows that the collection is an open cover of X. By compactness, there is a finite subcover, and since E_n are ascending the largest of these is a cover too. Thus we obtain that there is some positive integer N such that E_N=X. That is, if n>N and x is a point in X, then, as desired.