We prove constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0,1] into ℝ is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas.
"The pseudocompactness of [0,1] is equivalent to the uniform continuity theorem." J. Symbolic Logic 72 (4) 1379 - 1384, December 2007. https://doi.org/10.2178/jsl/1203350793