Question 6
Let $f: \mathbb{N} \to \mathbb{N}$ be a function satisfying
$$
k f(n) \le f(kn) \le k f(n) + k - 1
$$
for every $k, n \in \mathbb{N}$, where $\mathbb{N}$ is the set of all positive integers.
(1) Show that, for every $a, b \in \mathbb{N}$,
$$
f(a) + f(b) \le f(a + b) \le f(a) + f(b) + 1.
$$
(2) Show that, if $f$ satisfies
$$
f(2007n) \le 2007 f(n) + 2005
$$
for every $n \in \mathbb{N}$, then there exists $c \in \mathbb{N}$ such that $f(2007c) = 2007 f(c)$.
$$
k f(n) \le f(kn) \le k f(n) + k - 1
$$
for every $k, n \in \mathbb{N}$, where $\mathbb{N}$ is the set of all positive integers.
(1) Show that, for every $a, b \in \mathbb{N}$,
$$
f(a) + f(b) \le f(a + b) \le f(a) + f(b) + 1.
$$
(2) Show that, if $f$ satisfies
$$
f(2007n) \le 2007 f(n) + 2005
$$
for every $n \in \mathbb{N}$, then there exists $c \in \mathbb{N}$ such that $f(2007c) = 2007 f(c)$.