Dini's theorem explained

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.^[1]

Formal statement

If

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

Proof

Let

\varepsilon > 0 be given. For each n\in N , let g _{n=f-f} _{n} , and let E _{n} be the set of those x\in X such that g _{n(x)<\varepsilon} . Each g _{n} is continuous, and so each E _{n} is open (because each E _{n} is the preimage of the open set ( -inf ty, \varepsilon) under g _{n} , a continuous function). Since (f _{n)} _{n\inN} is monotonically increasing, (g _{n)} _{n\inN} is monotonically decreasing, it follows that the sequence E _{n} is ascending (i.e. E _{n\subset} E _{n+1} for all n\in N ). Since (f _{n)} _{n\inN} converges pointwise to f , it follows that the collection (E _{n)} _{n\inN} is an open cover of X . By compactness, there is a finite subcover, and since E _{n} are ascending the largest of these is a cover too. Thus 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 |f(x) - f _{n(x)|<\varepsilon} , 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.
• Book: Edwards , Charles Henry . Advanced Calculus of Several Variables . Dover Publications . Mineola, New York . 1994 . 1973 . 978-0-486-68336-2.
• Book: Graves , Lawrence Murray . The theory of functions of real variables . Dover Publications . Mineola, New York . 2009 . 1946 . 978-0-486-47434-2.
• Book: Friedman , Avner . Avner Friedman . Advanced calculus . Dover Publications . Mineola, New York . 2007 . 1971 . 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.
• Book: Thomson. Brian S.. Bruckner. Judith B.. Bruckner. Andrew M.. Andrew M. Bruckner. Elementary Real Analysis. 2008. 2001. ClassicalRealAnalysis.com. 978-1-4348-4367-8.

Notes and References

1. . . . .
2. According to, "[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".