Dini's theorem

In the mathematical field of analysis, Dini's theorem states that if X is a compact topological space, and { f_n } is a monotonically increasing sequence (meaning f_n( x ) ≤ f_n+1( x ) for all n and x) of continuous real-valued functions on X which converges pointwise to a continuous function f, then the convergence is uniform. The same conclusion holds if { f_n } is monotonically decreasing instead of increasing.

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. Note also that the limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous. (if the limit function is not required to be continuous, the theorem doesn't hold, as seen by the sequence f_n(x)=xⁿ over [0,1]).

Proof

Let ε > 0 be given. For each n, let g_n = f − f_n, and let E_n be those x ∈ X such that g_n( x ) < ε. Each g_n is continuous, and so each E_n is open. Since { f_n } is monotonically increasing, { g_n } is monotonically decreasing, it follows that the sequence E_n is ascending. Since f_n converges pointwise to f, it follows that the collection { E_n } is an open cover of X. By compactness, 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 )| < ε, as desired.

References

• Rudin, Walter R. (1976) Principles of Mathematical Analysis, Third Edition, McGraw-Hill. See Theorem 7.13 on page 150.
• Bartle, Robert G. and Sherbert Donald R.(2000) "Introduction to Real Analysis, Third Edition" Wiley. p 238.

-Presents a proof using gauges. Also, note Rudin only states the theorem for a decreasing sequence of functions.