Let \(Z_{rn}\) be the \(r\)th largest (\(r\geq 1\)) of \(\{X_1,X_2,\ldots,X_n\}\), where \(X_1,X_2,\ldots\) is a sequence of i.i.d. random variables with the distribution function \(F(x)\). The main result of the paper states that \(P\{Z_{rn}\leq u_n,\;\text{i.o.}\}=0\) or 1 accordingly as the series \[ \sum_{n=1}^\infty\exp\bigl[ -n\big\{ 1-F(u_n)\big\} \big] \big[ \big\{ 1-F(u_n)\big\}\big]^r /n<\infty\;\text{or} =\infty \] for any real sequence \(\{u_n\}\) such that \(\lim_{n\to\infty}n\{ 1-F(u_n)\}=+\infty\). In this proposition the condition imposed on the sequence \([n\{1-F(u_n)\}]\) is weaker than in [\textit{T. Mori}, Z. Wahrscheinlichkeitstheorie Verw. Geb. 36, 189-194 (1976; Zbl 0325.60033)]. This proposition generalizes to the case \(r\geq 1\) the results of \textit{M. J. Klass} [Ann. Probab. 12, 380-389 (1984; Zbl 0536.60038); ibid. 13, 1369-1370 (1985; Zbl 0576.60023)].