Question 23
Given a sequence $a_{1}, a_{2}, \ldots, a_{n}$ of real numbers. For each $i$ ($1 \leq i \leq n$) define
$$
d_{i}=\max \{a_{j}: 1 \leq j \leq i\}-\min \{a_{j}: i \leq j \leq n\}
$$
and let
$$
d=\max \{d_{i}: 1 \leq i \leq n\} .
$$
a. Prove that for arbitrary real numbers $x_{1} \leq x_{2} \leq \ldots \leq x_{n}$,
$$
\max \{\left|x_{i}-a_{i}\right|: 1 \leq i \leq n\} \geq \frac{d}{2} . \tag{1}
$$
b. Show that there exists a sequence $x_{1} \leq x_{2} \leq \ldots \leq x_{n}$ of real numbers such that we have equality in (1).
$$
d_{i}=\max \{a_{j}: 1 \leq j \leq i\}-\min \{a_{j}: i \leq j \leq n\}
$$
and let
$$
d=\max \{d_{i}: 1 \leq i \leq n\} .
$$
a. Prove that for arbitrary real numbers $x_{1} \leq x_{2} \leq \ldots \leq x_{n}$,
$$
\max \{\left|x_{i}-a_{i}\right|: 1 \leq i \leq n\} \geq \frac{d}{2} . \tag{1}
$$
b. Show that there exists a sequence $x_{1} \leq x_{2} \leq \ldots \leq x_{n}$ of real numbers such that we have equality in (1).