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 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.
[edit] Proof
Let 
be given. For each n, let gn = f − fn, and let En be those 
such that 
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 
as desired.
