Did Cauchy think that uniform and pointwise convergence were equivalent? 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.
 A: I recommend the reading of a wonderful 15-page text of Imre Lakatos, "Another case study of the method of proofs and refutations", which discuss precisely what Cauchy knew and didn't know and did and didn't do about this. It is published as an appendix in "Proofs and Refutations".
Google Book page of "Proofs and Refutations":
A: 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, including this 1987 publication in Historia Mathematica.
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 acknowledges that the condition as stated in 1821 needs to be modified.
Here Cauchy seems to introduce a condition close to uniform convergence which is stronger than one used in his earlier works. See for instance this 2013 publication in Foundations of Science and this 2018 publication in Mat.Stud..
A: 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."
