User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35392788-20191029194318/@comment-35470197-20191031220647

Great! If I am correct, the condition holds for typical fundamental sequences, especially for ordinal notations associated to standard OCFs, and implies the desired property, i.e. \(f_{\beta}\) eventually dominates \(f_{\alpha}\) for any \(\alpha < \beta). Although I do not know the original question, i.e. the relation between X(n) and E(n), even under this condition, it is very good.