Exercise 8, Appendix 0

a) Let $(f_n)$ be a sequence of increasing continuous functions belonging to $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ and such that $f_n \ll f_{n+1}$ for every $n$. Show that there exists a function $f$ that is continuous, increasing, belongs to $\mathscr{H}(\mathfrak{F}, \mathbf{R})$, and is such that $f \gg f_n$ for every $n$ (reduce to the case where $f_n \leq f_{n+1}$ and define $f$ so that $f_n(x) \leq f(x) \leq f_{n+1}(x)$ for $n \leq x \leq n+1$).

b) Let $(f_n)$ be a sequence of increasing continuous functions belonging to $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ and such that $1 \ll f_{n+1} \ll f_n$ for every $n$. Show that there exists a function $f$ that is continuous, increasing, belongs to $\mathscr{H}(\mathfrak{F}, \mathbf{R})$, and is such that $1 \ll f \ll f_n$ for every $n$ (reduce to the case where $f_{n+1} \leq f_n$ and show that one can define an increasing sequence $(x_n)$ of real numbers and a continuous increasing function $f$ such that $f_{n+1}(x) \leq f(x) \leq f_n(x)$ for $x_n \leq x \leq x_{n+1}$).

c) Let $(f_n), (g_n)$ be two sequences of continuous increasing functions belonging to $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ such that $f_n \ll f_{n+1}, g_m \gg g_{m+1}$ and $f_n \ll g_m$ for all $m$ and $n$; show that there exists a function $h$, continuous, increasing, belonging to $\mathscr{H}(\mathfrak{F}, \mathbf{R})$, such that $f_n \ll h \ll g_m$ for all $m$ and $n$ (similar method).

In particular, show that there exists a continuous decreasing function $f$ in $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ such that no logarithmic criterion allows one to determine whether the integral $\int_a^{+\infty} f(t)\,dt$ converges or is infinite ("*theorems of Du Bois-Reymond*"").