The infinite-sequence monotonicity determination claim

Let $a_n|_{n=d4a}$ be a sequence evaluated at infinity, and let $z$ denote a relation in $*G$. Infinite-sequence monotonicity claim. The monotonicity of $a_n|_{n=d4a}$ can be determined by solving for $z$ in

$$a_{n+1}\;z\;a_n|_{n=d4a},$$

or, when it exists, by using the continuous version

$$a(n+1)\;z\;a(n).$$

This is proposed as a method for determining monotonicity by comparing successive terms, extending the analogous derivative-based approach for continuous functions. The source gives no resolution.