The article by Fred Rickey that @viktorblasjo cites below is a pedagogical gem. Anyone teaching real analysis or advanced calculus should have a look. This is the right way to motivate the idea of uniform convergence — the way it arose historically, from Cauchy’s fruitful error.https://twitter.com/viktorblasjo/status/1167164076339974145 …
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Love Lakatos. Actually I think his take on Cauchy is different than Rickey implies. According to Lakatos, the counterexample Fourier series are not convergent to Cauchy, because they don't converge at some "point" such as x=c+1/n due to Gibbs phenomenon bump that doesn't go away.
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So it's not that Cauchy made hidden assumptions without realising it, or didn't realise the distinction between uniform and non-uniform convergence. Rather, Cauchy precisely *did* see this distinction and decided that non-uniform convergence is not convergence at all!
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