I've heard that Cauchy thought he'd proved that pointwise and uniform convergence are equivalent. Is this a historical fact? If it is indeed true, I was wondering if anyone had a reference.

See the wonderful book by Judith Grabiner, The Origins of Cauchy's Rigorous Calculus (1981), page 140 : "Actually, his [Cauchy's] proof implicitly assumed the function to be uniformly continuous, though he did not distinguish between continuity and uniform continuity, just as he had not distinguished between convergence and uniform convergence." 


The issue of Cauchy's understanding of continuity is a subject of lively historical debate. Grabiner represents only one view in this debate. Laugwitz has published a series of scholarly articles studying the issue. No discussion of this issue is complete without mentioning Cauchy's article dating from 1853 where he deals with convergence of series of functions, and seems to introduce a condition close to uniform convergence which is stronger than one used in his earlier works. See for instance this article. 

