Shouldn't the theorem state that the sequence of partial sums converges uniformly on $S$?
Cauchy criterion uniformly on $S\implies$ sequence converges uniformly on $S$
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uniform-convergence
cauchy-sequences
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0Can I ask from what book is this? – 2017-02-05
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1@Masacroso Elementary Analysis: The Theory of Calculus by Kenneth A. Ross – 2017-02-05
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3"the series converges" = "the sequence of partial sums converges". That's the definition of "convergence of a series". – 2017-02-05
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0As Sassatelli said when one is talking of a convergent series then it holds that $$\sum_{k=0}^\infty a_k:=\lim_{n\to\infty}\sum_{k=0}^n a_k=c$$ where the LHS is a notation that represent the limit of the RHS. – 2017-02-05