Question 87
Let $N = \{0, 1, 2, 3, \dots\}$. Given sets $A, B \subset N$, for each positive integer $n$ denote $r(A, B, n)$ as the number of solutions to the equation $a + b = n$, $a \in A, b \in B$.
Prove that there exists $n_0 \in N$ such that $r(A, B, n+1) \gt r(A, B, n)$ for all $n \gt n_0$ if and only if $N \setminus A$ and $N \setminus B$ are both finite.
Prove that there exists $n_0 \in N$ such that $r(A, B, n+1) \gt r(A, B, n)$ for all $n \gt n_0$ if and only if $N \setminus A$ and $N \setminus B$ are both finite.