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Part of 2003 Miklós Schweitzer

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Miklós Schweitzer 2003, Problem 8

Source: Miklós Schweitzer

7/30/2016
Let f1,f2,…f_1, f_2, \ldots be continuous real functions on the real line. Is it true that if the series ∑n=1∞fn(x)\sum_{n=1}^{\infty} f_n(x) is divergent for every xx, then this holds also true for any typical choice of the signs in the sum (i.e. the set of those {ϵn}n=1∞∈{+1,−1}N\{ \epsilon _n\}_{n=1}^{\infty} \in \{ +1, -1\}^{\mathbb{N}} sequences, for which there series ∑n=1∞ϵnfn(x)\sum_{n=1}^{\infty} \epsilon_nf_n(x) is convergent at least at one point xx, forms a subset of first category within the set {+1,−1}N\{+1,-1\}^{\mathbb{N}} )?
(translated by L. Erdős)
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