In the mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 field of analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, Dini's theorem states that if X is a compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

, and { f_n } is a monotonically increasing sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

 (meaning f_n( x ) ≤ f_n+1( x ) for all n and x) of continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 real-valued function

Real-valued function

In mathematics, a real-valued function is a function that associates to every element of the domain a real number in the image....

s on X which converges pointwise

Pointwise convergence

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...

 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.