Dini's Theorem states that if a sequence of continuous functions on a compact space converges pointwise and monotonically to a continuous limit function, then the convergence is in fact uniform. Named for Ulisse Dini, it is a standard result of real analysis identifying one of the more common conditions under which pointwise convergence upgrades to the stronger, more useful property of uniform convergence.
Facts
StatementIf a monotone sequence of continuous functions converges pointwise on a compact space and the limit function is also continuous, then the convergence is uniform. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Dini's Theorem (Wikipedia)
Wikimedia FoundationLead section, theorem statement sentence
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.
Lead section, naming and history sentence
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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