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.^[1]

Formal statement

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

Proof

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

Notes

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.