Exercise 7, Appendix 0

Let $f$ be an (H) function having a definition sequence of $n$ terms. Show that if one has $e_p((l_{q+1}(x))^\beta) \ll f(x) \ll e_p((l_q(x))^\alpha)$ (resp. $e_p((l_q(x))^\beta) \ll f(x) \ll e_{p+1}((l_q(x))^\alpha)$) for every $\beta > 0$ and every $\alpha$ such that $0 < \alpha < 1$, then necessarily $p + q + l < n$ (argue by induction on $n$, using methods similar to those of exerc. 1 (V, p. 263) and 4 (V, p. 264)).