In the present paper we give very simple general statements which deal with approximation of a real number by rationals and are related to isolation phenomenon. In particular we study functions $ f(x)>f_1(x)>0$ such that existence of solutions $\frac{ p}{ q}$ of Diophantine inequality $ \left| \alpha -\frac{ p}{ q}\right|< \frac{ f(q)}{ q^2} $ leads to the existence of solutions of inequality $ \left| \alpha -\frac{ p}{ q}\right|< \frac{ f_1(q)}{ q^2} $.
Categories: math.NT