Question 1
Real numbers $a_1$, $a_2$, …, $a_n$ are fixed. For each $1 \le i \le n$ we let $d_i = \max\{a_j : 1 \le j \le i\} - \min\{a_j : i \le j \le n\}$ and let $d = \max \{d_i : 1 \le i \le n\}$.
(a) Prove that for any real numbers $x_1 \le \dots \le x_n$ we have $$
\max \left\{ \left\lvert x_i - a_i \right\rvert :
1 \le i \le n \right\}
\ge \frac12 d.
$$
(b) Moreover, show that there exists some choice of $x_1 \le \dots \le x_n$ which achieves equality.
(a) Prove that for any real numbers $x_1 \le \dots \le x_n$ we have $$
\max \left\{ \left\lvert x_i - a_i \right\rvert :
1 \le i \le n \right\}
\ge \frac12 d.
$$
(b) Moreover, show that there exists some choice of $x_1 \le \dots \le x_n$ which achieves equality.