Working on a proof right now and would like to use the following reasoning:
Given a set $R \subset \Bbb N \times\Bbb N$ and a number $b \in \Bbb N$. If we know that there is a subset $R_1 \subset R$ such that
$$\sum_{(a_j, a_k) \in R_1} a_j \le b \ \textit{and} \sum_{(a_j, a_k) \in R_1} a_k \ge b,$$
is it valid to reason further that there must be elements $a_1, \ ... \ ,a_n$ (that are part of the first components of elements of $R_1$ or of the second compononents of elements of $R_1$) such that
$$\sum_{a_i} a_i = b?$$