On certain approximation problem connected with the sums of subseries (Q2843901)

The authors consider a problem of approximating real numbers by using some families of real numbers satisfying some conditions. We say that a nonempty set \(w\subset s=\{\{a_n\}: a_n\geq 0 \, \text{and} \sum a_n<\infty \}\) is closed under initial subsequences (cuis-set) if from the fact that for a given sequence \(\{a_n\}\in s\) and for any \(N\in \mathbb N\) there exists \(\{b_n\}\in w\) such that \(a_n=b_n\) for \(n=1,2,\dots, N\), it follows that \(\{a_n\}\in w\).NEWLINENEWLINEA typical result is the following theorem. Let \(w\subset s\) be a cuis-set. Let \(\{a_n\}\in s\) and \(x>0\). If for every \(\epsilon >0\) there exists a subsequence \(\{a_{n_i}\}\in w\) such that \(x-\epsilon <\sum a_{n_i}<x\), then there also exists a subsequence \(\{a_{n_i}\}\in w\) such that \(\sum a_{n_i}=x\).