Dini's theorem

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

Proof

Let $\varepsilon >0$ be given. For each $n\in \mathbb {N}$ , 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 $(-\infty ,\varepsilon )$ under $g_{n}$ , a continuous function). Since $(f_{n})_{n\in \mathbb {N} }$ is monotonically increasing, $(g_{n})_{n\in \mathbb {N} }$ is monotonically decreasing, it follows that the sequence $E_{n}$ is ascending (i.e. $E_{n}\subset E_{n+1}$ for all $n\in \mathbb {N}$ ). Since $(f_{n})_{n\in \mathbb {N} }$ converges pointwise to $f$ , it follows that the collection $(E_{n})_{n\in \mathbb {N} }$ 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.

Notes

^ Edwards 1994, p. 165. Friedman 2007, p. 199. Graves 2009, p. 121. Thomson, Bruckner & Bruckner 2008, p. 385.
^ 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".

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) [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.

[1] Edwards 1994, p. 165. Friedman 2007, p. 199. Graves 2009, p. 121. Thomson, Bruckner & Bruckner 2008, p. 385.

[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".

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