Max Before, Min After

Let $\{a_n\}$ be an infinite sequence of nonnegative integers. Let $A_n$ denote the maximum of the first $n$ terms, and $B_n$ the minimum of the terms after the $n$th: $B_n = \min\{a_{n+1}, a_{n+2}, \dots\}$. Set $d_n = A_n - B_n$.

1. If $\{a_n\}$ is $2, 1, 4, 3, 2, 1, 4, 3, \dots$, periodic with period $4$ (that is, $a_{n+4} = a_n$ for all $n$), write down $d_1, d_2, d_3, d_4$. 2. Let $d$ be a nonnegative integer. Prove that $d_n = -d$ for all $n$ if and only if $\{a_n\}$ is an arithmetic sequence with common difference $d$. 3. Prove that if $a_1 = 2$ and $d_n = 1$ for all $n$, then every term of $\{a_n\}$ equals $1$ or $2$, and infinitely many terms equal $1$.