Let us have a sequence $a_n$ of positive rational numbers, for which $\sum a_n = R \in \mathbb R$. Now suppose $b_n$ is a subsequence of $a_n$ such that $\sum b_n = r < R$. The question is "Can there be an uncountable number of subsequences of $a_n$ whose sum is $r$, for some sequence of positive rationals $a_n$ and some $r\in\mathbb R$?"
To be clear on definitions, a sequence and all its subsequences must be mappings from $\mathbb N\to X$, where in this case $X$ is the set of positive rational numbers.
Best Answer
Yes, consider this series:
$$1+\frac12+\frac12+1+\frac14+\frac18+\frac18+\frac14+\frac1{16}+\frac1{32}+\frac1{32}+\frac1{16}+\cdots=4$$
There are uncountably many subsequences that sum to $2$. From each quadruplet of the form $2a+a+a+2a$, we may select either $2a+a$ or $a+2a$.