We study increasing $F$-sequences, where $F$ is a dilator: an increasing $F$-sequence is a sequence (indexed by ordinal numbers) of ordinal numbers, starting with 0 and terminating at the first step $x$ where $F(x)$ is reached (at every step $x + 1$ we use the same process as in decreasing $F$-sequences, cf. [2], but with "+ 1" instead of "- 1"). By induction on dilators, we shall prove that every increasing $F$-sequence terminates and moreover we can determine for every dilator $F$ the point where the increasing $F$-sequence terminates. We apply these results to inverse Goodstein sequences, i.e. increasing $(1 + Id)_{(\omega)}$-sequences. We show that the theorem every inverse Goodstein sequence terminates (a combinatorial theorem about ordinal numbers) is not provable in $ID_1$. For a general presentation of the results stated in this paper, see [1]. We use notions and results concerning the category ON (ordinal numbers), dilators and bilators, summarized in [2, pp. 25-31].