- Dini's theorem
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In the mathematical field of analysis, Dini's theorem states that 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.
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. (if the limit function is not required to be continuous, the theorem doesn't hold, as seen by the sequence fn(x)=xn over [0,1]).
Proof
Let ε > 0 be given. For each n, let gn = f − fn, and let En be those x ∈ X such that gn( x ) < ε. Each gn is continuous, and so each En is open. 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.
References
- Rudin, Walter R. (1976) Principles of Mathematical Analysis, Third Edition, McGraw-Hill. See Theorem 7.13 on page 150.
- Bartle, Robert G. and Sherbert Donald R.(2000) "Introduction to Real Analysis, Third Edition" Wiley. p 238.
-Presents a proof using gauges. Also, note Rudin only states the theorem for a decreasing sequence of functions.
Categories:- Theorems in real analysis
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