On certain approximation problem connected with the sums of subseries

Abstract

In this paper a problem of approximating
the real numbers by using the series of real numbers is considered.
It is proven that if the given family of sequences of real numbers satisfies
some conditions of set-theoretical nature, like being closed under initial subsequences
and (additionally) possessing properties of adding and removing elements, then  it automatically possesses
some approximating properties, like, for example, reaching supremum of the set of sums of subseries.