Consider a sequence $x_n$ with positive values such that $\sum _{n=1}^{\infty} x_n $ converges . Is the set of the subsequences $ x_{k_n}$ of $x_n$ such that $\sum _{n=1}^{\infty} x_{k_n} =c \in R$ countable ? $c$ is previously fixed so the corresponding series of every subsequence converges to the same value.
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I have answered assuming that you mean the set of subsequences such that the corresponding series converges. Do you mean that $c$ is previoulsy fixed instead? That is, your question is that, given $c\in \Bbb R$, the set of subsequences such that $\sum x_{k_n}=c$ is countable? – ajotatxe Dec 11 '18 at 17:53
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Yes I mean that c if previously fixed .So for all the subsequences their corresponding series converges to the same c in $R$ – mike moke Dec 11 '18 at 17:57
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Then the answer depends clearly on $c$. For example, if $c>\sum x_n$, the set is void. If $c=\sum x_n$ the set is a singleton. For intermediate values, it depends also on the terms of the sequence. – ajotatxe Dec 11 '18 at 19:13
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In both cases the set is countable .If the answer is no then we need to construct an uncountable set . – mike moke Dec 11 '18 at 19:17