Question 160
We say that a set $B$ of integers is an integer interval if there exist integers $i \leqslant j$ such that $B=\{i, i+1, \ldots, j\}$, and we denote by $\mathcal{J}$ the set of integer intervals. Moreover, if $A=\{a_1, \ldots, a_k\}$ is a set of integers, with $a_1 \lt \ldots \lt a_k$, we set
$$
f(A)=\max_{1 \leqslant i \lt k}(a_{i+1}-a_i) \text{ and } g(A)=\max_{B \subseteq A, B \in \mathcal{J}} |B|.
$$
Recall that the notation $|B|$ denotes the number of elements of the set $B$. If $A$ is empty or a singleton, we also set $f(A)=0$. Finally, for every integer $n \geqslant 1$, we define
$$
F(n)=\sum_{A \subseteq\{1, \ldots, n\}} f(A) \text{ and } G(n)=\sum_{A \subseteq\{1, \ldots, n\}} g(A).
$$
Show that there exists a natural number $m$ such that $F(n) \gt G(n)$ for every integer $n \geqslant m$.
$$
f(A)=\max_{1 \leqslant i \lt k}(a_{i+1}-a_i) \text{ and } g(A)=\max_{B \subseteq A, B \in \mathcal{J}} |B|.
$$
Recall that the notation $|B|$ denotes the number of elements of the set $B$. If $A$ is empty or a singleton, we also set $f(A)=0$. Finally, for every integer $n \geqslant 1$, we define
$$
F(n)=\sum_{A \subseteq\{1, \ldots, n\}} f(A) \text{ and } G(n)=\sum_{A \subseteq\{1, \ldots, n\}} g(A).
$$
Show that there exists a natural number $m$ such that $F(n) \gt G(n)$ for every integer $n \geqslant m$.