Dini's theorem says that compactness of the domain, a metric space, ensures the uniform convergence of every simply convergent monotone sequence of real-valued continuous functions whose limit is continuous. By showing that Dini's theorem is equivalent to Brouwer's fan theorem for detachable bars, we provide Dini's theorem with a classification in the recently established constructive reverse mathematics propagated by Ishihara. As a complement, Dini's theorem is proved to be equivalent to the analogue of the fan theorem, weak König's lemma, in the original classical setting of reverse mathematics started by Friedman and Simpson.
"Classifying Dini's Theorem." Notre Dame J. Formal Logic 47 (2) 253 - 262, 2006. https://doi.org/10.1305/ndjfl/1153858650