Some properties of rapidly varying sequences

We investigate the class R∞,s of rapidly varying sequences from different points of view. It is shown: (1) a sequence (cn)n∈N of positive real numbers is rapidly varying if and only if the corresponding function x→c[x], x⩾1, is rapidly varying; (2) the class R∞,s satisfies some selection properties, as well as game-theoretical and Ramsey-theoretical conditions.