Fixing $n\in\Bbb N$, there are infinitely many $r\in\Bbb N\smallsetminus\{n\}$, and so infinitely many sets $\{n,r\}$ with $r\in\Bbb N\smallsetminus\{n\}$. Now, each $\{n,r\}$ is mapped to either $1$ or $2$. If there were only finitely many $\{n,r\}$ mapping to $1$ and only finitely many $\{n,r\}$ mapping to $2$, then since a union of two finite sets is finite, we would only have finitely many $\{n,r\}$ to begin with, which we've already seen is not the case. Hence, there are infinitely many $\{n,r\}$ mapping to $1$ or there are infinitely many $\{n,r\}$ mapping to $2$.
If there are infinitely many $r$ such that $f$ maps $\{n,r\}\mapsto 1$, then we let $R_1=\left\{r\in\Bbb N:\{n,r\}\in f^{-1}\bigl(\{1\}\bigr)\right\}.$ Otherwise, we let $R_1=\left\{r\in\Bbb N:\{n,r\}\in f^{-1}\bigl(\{2\}\bigr)\right\}.$