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In-Depth Information
During the eighteenth century it was presumed that the limit of a
sequence of continuous functions was continuous. In 1821, Cauchy
believed that he had a proof of this and in 1823 a proof that a sequence
of integrable functions was integrable. However, he had not identified
the distinctive features of uniform convergence and his proofs were
fallacious. This was pointed out by Abel (1826) using an example due to
Fourier. Abel showed that a power series was uniformly convergent
inside its radius of convergence and showed that this convergence
implied the continuity of the limit. When Abel reflected on why the
results in analysis obtained before Cauchy were generally sound, despite
the lack of precision relating to limiting processes, he suggested that the
reason might be that the functions considered were expressible as power
series. In the same paper, Abel proved the binomial theorem for
complex numbers, stimulated by a paper by Bolzano (1816) which
exposed the flaws in earlier treatments and Cauchy's treatment for real
index (1821). In 1848 Seidel correctly analysed the defect in Cauchy's
proofs, but his discussion of uniform convergence was clumsy by
comparison with that of Weierstrass in his lectures in Berlin in the
1860s, to whom we owe the theorems of this chapter. Weierstrass was
able to prove that every continuous function was the uniform limit of a
sequence of polynomials and believed that this theorem legitimated the
study of continuous but non-differentiable functions which he had
initiated after Riemann's definition of the integral.
