Question 166
We say a function $f: \mathbb{Z}_{\ge 0} \times \mathbb{Z}_{\ge 0} \to \mathbb{Z}$ is *great* if for any nonnegative integers $m$ and $n$,
$$
f(m+1, n+1)f(m, n) - f(m+1, n)f(m, n+1) = 1.
$$
If $A = (a_0, a_1, \dots)$ and $B = (b_0, b_1, \dots)$ are two sequences of integers, we write $A \sim B$ if there exists a great function $f$ satisfying $f(n, 0) = a_n$ and $f(0, n) = b_n$ for every nonnegative integer $n$ (in particular, $a_0 = b_0$).
Prove that if $A, B, C$, and $D$ are four sequences of integers satisfying $A \sim B, B \sim C$, and $C \sim D$, then $D \sim A$.
$$
f(m+1, n+1)f(m, n) - f(m+1, n)f(m, n+1) = 1.
$$
If $A = (a_0, a_1, \dots)$ and $B = (b_0, b_1, \dots)$ are two sequences of integers, we write $A \sim B$ if there exists a great function $f$ satisfying $f(n, 0) = a_n$ and $f(0, n) = b_n$ for every nonnegative integer $n$ (in particular, $a_0 = b_0$).
Prove that if $A, B, C$, and $D$ are four sequences of integers satisfying $A \sim B, B \sim C$, and $C \sim D$, then $D \sim A$.