Non-classical study on the simultaneous rational approximation

This study is placed in the framework of internal set theory (nelson,1977). real numbers (ξi) i=1,2,...,k are called simultaneously approximable in the infinitesimal sense,if for every positive infinitesimal ε,there exist rational numbers (pi/q) i=1,2,...,k such that where (£i) i=1,2,...,k are limited numbers. let (ξ0,ξ1,...,ξε) be a system of reals,with ε unlimited. in this paper,we will give a necessary condition for which (ξi) i=1,2,...,k are simultaneously approximable in the infinitesimal sense. the converse of this condition is also discussed.