Question 16
A subset $B$ of natural numbers is called *loyal* if there exist positive integers $i \le j$ such that $B = \{i, i+1, \dots, j\}$. $Q$ is the collection of *loyal* subsets of natural numbers. For every subset $A = \{a_1 \lt a_2 \lt \dots \lt a_k\}$ of $\{1, 2, \dots, n\}$ we define:
$$
g(A) = \max_{B \subset A, B \in Q} |B| \quad \text{and} \quad f(A) = \max_{1 \le i \le k-1} a_{i+1} - a_i.
$$
Also, Define
$$
G(n) = \sum_{A \subseteq \{1, 2, \dots, n\}} g(A) \quad \text{and} \quad F(n) = \sum_{A \subseteq \{1, 2, \dots, n\}} f(A).
$$
Prove there exists $m \in \mathbb{N}$ such that for every positive integer $n$ greater than $m$ we have $F(n) \gt G(n)$. ($|A|$ denotes the number of elements of a set $A$ and if $|A| \le 1$ we set $f(A) = 0$).
$$
g(A) = \max_{B \subset A, B \in Q} |B| \quad \text{and} \quad f(A) = \max_{1 \le i \le k-1} a_{i+1} - a_i.
$$
Also, Define
$$
G(n) = \sum_{A \subseteq \{1, 2, \dots, n\}} g(A) \quad \text{and} \quad F(n) = \sum_{A \subseteq \{1, 2, \dots, n\}} f(A).
$$
Prove there exists $m \in \mathbb{N}$ such that for every positive integer $n$ greater than $m$ we have $F(n) \gt G(n)$. ($|A|$ denotes the number of elements of a set $A$ and if $|A| \le 1$ we set $f(A) = 0$).