• Images
• Videos
• Documents
• Books
• Articles

Formal statement

If X {\displaystyle X} is a compact topological space, and ( f n ) n ∈ N {\displaystyle (f_{n})_{n\in \mathbb {N} }} is a monotonically increasing sequence (meaning f n ( x ) ≤ f n + 1 ( x ) {\displaystyle f_{n}(x)\leq f_{n+1}(x)} for all n ∈ N {\displaystyle n\in \mathbb {N} } and x ∈ X {\displaystyle x\in X} ) of continuous real-valued functions on X {\displaystyle X} which converges pointwise to a continuous function f : X → R {\displaystyle f\colon X\to \mathbb {R} } , then the convergence is uniform. The same conclusion holds if ( f n ) n ∈ N {\displaystyle (f_{n})_{n\in \mathbb {N} }} is monotonically decreasing instead of increasing. The theorem is named after Ulisse Dini.²

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 x n {\displaystyle x^{n}} in [ 0 , 1 ] {\displaystyle [0,1]} .)

Proof

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

Notes

• 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) [1973]. Advanced Calculus of Several Variables. Mineola, New York: Dover Publications. ISBN 978-0-486-68336-2.
• Graves, Lawrence Murray (2009) [1946]. The theory of functions of real variables. Mineola, New York: Dover Publications. ISBN 978-0-486-47434-2.
• Friedman, Avner (2007) [1971]. 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) [2001]. Elementary Real Analysis. ClassicalRealAnalysis.com. ISBN 978-1-4348-4367-8.

References

1. Edwards 1994, p. 165. Friedman 2007, p. 199. Graves 2009, p. 121. Thomson, Bruckner & Bruckner 2008, p. 385. - Edwards, Charles Henry (1994) [1973]. Advanced Calculus of Several Variables. Mineola, New York: Dover Publications. ISBN 978-0-486-68336-2. ↩

2. According to Edwards 1994, p. 165, "[This theorem] is called Dini's theorem because Ulisse Dini (1845–1918) presented the original version of it in his book on the theory of functions of a real variable, published in Pisa in 1878". - Edwards, Charles Henry (1994) [1973]. Advanced Calculus of Several Variables. Mineola, New York: Dover Publications. ISBN 978-0-486-68336-2. ↩