Question 5
Let $n$ be a positive integer and $\mathbb{R}^n$ be the set of ordered $n$-tuples of real numbers. Let $T$ denote the collection of $(x_1, x_2, \dots, x_n) \in \mathbb{R}^n$ for which there exists a permutation $\sigma$ of $1, 2, \dots, n$ such that $x_{\sigma(i)} - x_{\sigma(i+1)} \ge 1$ for each $1 \le i \lt n$. Prove that there is a real number $d$ satisfying the following condition:
For every $(a_1, a_2, \dots, a_n) \in \mathbb{R}^n$, there exist $(b_1, b_2, \dots, b_n)$, $(c_1, c_2, \dots, c_n) \in T$ such that
$$
a_i = \frac{1}{2}(b_i + c_i), \quad |a_i - b_i| \le d, \quad |a_i - c_i| \le d \quad (1 \le i \le n).
$$
For every $(a_1, a_2, \dots, a_n) \in \mathbb{R}^n$, there exist $(b_1, b_2, \dots, b_n)$, $(c_1, c_2, \dots, c_n) \in T$ such that
$$
a_i = \frac{1}{2}(b_i + c_i), \quad |a_i - b_i| \le d, \quad |a_i - c_i| \le d \quad (1 \le i \le n).
$$