Exercise 1, § 4

a) Let $(x_n, y_n)_{n \in \mathbf{Z}}$ be a family of points of $\mathbf{R}^2$ such that for all $n \in \mathbf{Z}$ we have $x_n < x_{n+1}$ and

Show that, for all integers $n \in \mathbf{Z}, m > 0, p > 0$, we have

b) If $a > 0$ and $x \neq 0$, put

Show that, if $x < y, x \neq 0, y \neq 0$ and $a \neq 1$, we have $f_a(x) < f_a(y)$ [prove the relation first for $x, y$ integral, by means of $a$], then for $x, y$ rational, and finally in general].

c) Deduce that, for all $a > 0$, the function $f_a(x)$, defined for $x \neq 0$, has a limit on the right and a limit on the left as $x \to 0$; show that these two limits have a common value $\varphi(a)$, and that $\varphi(a) \neq 0$ if $a \neq 1$. Show that if $a \neq 1$, the function $\log_a x/x - 1$, defined for $x \neq 1$, tends to the limit $1/\varphi(a)$ as $x \to 1$.

d) Show that, for all $a > 0$ and $b > 0$, we have $\varphi(b) = \varphi(a) \log_a b$ (if $a \neq 1$). Deduce that there exists a number $e$ between 2 and 4 such that $\varphi(e) = 1$, and that $\varphi(a) = \log_e a$ for all $a > 0$.