Question 9
Let $n \geqslant 3$ be an integer and let $a_{1}, a_{2}, \ldots, a_{n}$ be real numbers.
a. Suppose that $a_{i} \lt \max \left(a_{i-1}, a_{i+1}\right)$ for every $i \in\{2,3, \ldots, n-1\}$. Show that $a_{i} \lt \max \left(a_{1}, a_{n}\right)$ for every $i \in\{2,3, \ldots, n-1\}$.
b. Suppose that $a_{i} \leqslant \max \left(a_{i-1}, a_{i+1}\right)$ for every $i \in\{2,3, \ldots, n-1\}$. Is it true that $a_{i} \leqslant \max \left(a_{1}, a_{n}\right)$ for every $i \in\{2,3, \ldots, n-1\}$?
N.B. If $x$ and $y$ are two real numbers, $\max (x, y)$ denotes the larger of the two.
a. Suppose that $a_{i} \lt \max \left(a_{i-1}, a_{i+1}\right)$ for every $i \in\{2,3, \ldots, n-1\}$. Show that $a_{i} \lt \max \left(a_{1}, a_{n}\right)$ for every $i \in\{2,3, \ldots, n-1\}$.
b. Suppose that $a_{i} \leqslant \max \left(a_{i-1}, a_{i+1}\right)$ for every $i \in\{2,3, \ldots, n-1\}$. Is it true that $a_{i} \leqslant \max \left(a_{1}, a_{n}\right)$ for every $i \in\{2,3, \ldots, n-1\}$?
N.B. If $x$ and $y$ are two real numbers, $\max (x, y)$ denotes the larger of the two.