关于利用递归定理证明可数非有限无最大元有向集支配序列存在性的问题咨询

Hey folks! I'm working through a problem using the Recursion Theorem and I've hit a couple of snags—two questions I'm hoping to get help with.

First, let me state the theorem I'm working with:

Theorem: For any class function $F:V\to V$, there exists a unique class function $G:ON\to V$ such that $(\forall\alpha):G(\alpha)=F((G(\beta):\beta<\alpha))$.

A quick note on the notation here: $V$ is the universal class defined as $V=\left{x\mid x=x\right}$, and $ON$ denotes the class of all ordinals.

Now, the problem I'm trying to solve using this theorem is:

Any countable, infinite ($|A|=\aleph_0$) directed set $A$ with no maximal element has a dominating sequence $\left\langle x_{n}\mid n\in\mathbb N{}\right\rangle$, such that for any $n\in \mathbb N$: [原文此处内容截断]

备注：内容来源于stack exchange，提问作者Gardosh