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 $X$ 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.

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.

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.