Dini's theorem

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 [math]\displaystyle{ X }[/math] is a compact topological space, and [math]\displaystyle{ (f_n)_{n\in\mathbb{N}} }[/math] is a monotonically increasing sequence (meaning [math]\displaystyle{ f_n(x)\leq f_{n+1}(x) }[/math] for all [math]\displaystyle{ n\in\mathbb{N} }[/math] and [math]\displaystyle{ x\in X }[/math]) of continuous real-valued functions on [math]\displaystyle{ X }[/math] which converges pointwise to a continuous function [math]\displaystyle{ f\colon X\to \mathbb{R} }[/math], then the convergence is uniform. The same conclusion holds if [math]\displaystyle{ (f_n)_{n\in\mathbb{N}} }[/math] 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 [math]\displaystyle{ x^n }[/math] in [math]\displaystyle{ [0,1] }[/math].)

Proof

Let [math]\displaystyle{ \varepsilon \gt 0 }[/math] be given. For each [math]\displaystyle{ n\in\mathbb{N} }[/math], let [math]\displaystyle{ g_n=f-f_n }[/math], and let [math]\displaystyle{ E_n }[/math] be the set of those [math]\displaystyle{ x\in X }[/math] such that [math]\displaystyle{ g_n(x)\lt \varepsilon }[/math]. Each [math]\displaystyle{ g_n }[/math] is continuous, and so each [math]\displaystyle{ E_n }[/math] is open (because each [math]\displaystyle{ E_n }[/math] is the preimage of the open set [math]\displaystyle{ (-\infty, \varepsilon) }[/math] under [math]\displaystyle{ g_n }[/math], a continuous function). Since [math]\displaystyle{ (f_n)_{n\in\mathbb{N}} }[/math] is monotonically increasing, [math]\displaystyle{ (g_n)_{n\in\mathbb{N}} }[/math] is monotonically decreasing, it follows that the sequence [math]\displaystyle{ E_n }[/math] is ascending (i.e. [math]\displaystyle{ E_n\subset E_{n+1} }[/math] for all [math]\displaystyle{ n\in\mathbb{N} }[/math]). Since [math]\displaystyle{ (f_n)_{n\in\mathbb{N}} }[/math] converges pointwise to [math]\displaystyle{ f }[/math], it follows that the collection [math]\displaystyle{ (E_n)_{n\in\mathbb{N}} }[/math] is an open cover of [math]\displaystyle{ X }[/math]. By compactness, there is a finite subcover, and since [math]\displaystyle{ E_n }[/math] are ascending the largest of these is a cover too. Thus we obtain that there is some positive integer [math]\displaystyle{ N }[/math] such that [math]\displaystyle{ E_N=X }[/math]. That is, if [math]\displaystyle{ n\gt N }[/math] and [math]\displaystyle{ x }[/math] is a point in [math]\displaystyle{ X }[/math], then [math]\displaystyle{ |f(x)-f_n(x)|\lt \varepsilon }[/math], as desired.

References

• Bartle, Robert G. and Sherbert Donald R.(2000) "Introduction to Real Analysis, Third Edition" Wiley. p 238. – Presents a proof using gauges.
• Edwards, Charles Henry (1994). Advanced Calculus of Several Variables. Mineola, New York: Dover Publications. ISBN 978-0-486-68336-2. 
• Graves, Lawrence Murray (2009). The theory of functions of real variables. Mineola, New York: Dover Publications. ISBN 978-0-486-47434-2. 
• Friedman, Avner (2007). Advanced calculus. Mineola, New York: Dover Publications. ISBN 978-0-486-45795-6. 
• Jost, Jürgen (2005) Postmodern Analysis, Third Edition, Springer. See Theorem 12.1 on page 157 for the monotone increasing case.
• Rudin, Walter R. (1976) Principles of Mathematical Analysis, Third Edition, McGraw–Hill. See Theorem 7.13 on page 150 for the monotone decreasing case.
• Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008). Elementary Real Analysis. ClassicalRealAnalysis.com. ISBN 978-1-4348-4367-8.