Dini's theorem

From Wikipedia, the free encyclopedia

Dini's theorem states that if X is a compact topological space, and {f_n) 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.

An analogous statement holds if {f_n} is monotonically decreasing.

[edit] Proof

Let be given. For each n, let g_n = f − f_n, and let E_n be those such that Plainly, each g_n is continuous, whence each E_n is open. Since {f_n} is monotonically increasing, {g_n} is monotonically decreasing, it follows swiftly that the sequence E_n is ascending. Since f_n converges pointwise to f, it follows that the collection (E_n} is an open cover of X. By compactness, 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.