Question 13
Let $(x_{n}),\ n \in \mathbb{N}$, a sequence of integers such that $$
\begin{aligned}
x_{1} & = 1 \\
x_{n+1} & \gt x_{n}, \text{ to } n \geq 1 \\
x_{n+1} & \leq 2n, \text{ to } n \geq 1
\end{aligned}
$$ demonstrate that for all natural integers $k$ there are two terms of the sequence $x_{r}$ and $x_{s}$ such that $x_{r} - x_{s} = k$.
\begin{aligned}
x_{1} & = 1 \\
x_{n+1} & \gt x_{n}, \text{ to } n \geq 1 \\
x_{n+1} & \leq 2n, \text{ to } n \geq 1
\end{aligned}
$$ demonstrate that for all natural integers $k$ there are two terms of the sequence $x_{r}$ and $x_{s}$ such that $x_{r} - x_{s} = k$.