Dini's theorem

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Dini's theorem states that if X is a compact topological space, and {fn} is a monotonically increasing sequence (meaning \scriptstyle f_n(x) \leq f_{n+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.

An analogous statement holds if {fn} is monotonically decreasing. 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, as shown by the counterexample {xn} on the interval [0,1], where n ranges over all positive integers. Each xn will map this interval to itself, it is easy to see that this sequence converges pointwise. Clearly the limit will be identically zero except at 1, where it takes the value 1. But the convergence is not uniform, because each y in [0,1] has an nth root for all positive integers n.

[edit] Proof

Let \varepsilon > 0 be given. For each n, let gn = f − fn, and let En be those x \in X such that g_n(x) < \varepsilon. Plainly, each gn is continuous, whence each En is open. Since {fn} is monotonically increasing, {gn} is monotonically decreasing, it follows swiftly 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) - f_n(x)| < \varepsilon, as desired.