Why allowed to use Axiom of Completeness in Archimedean principle?
The axiom of completeness is a statement about real numbers, but in the proof of theorem 1.4.2, Abbott says "assume $\mathbb N$ is bounded above, then by AoC, $\mathbb{N}$ should have a least upper bound", but AoC applies only to $\mathbb R$. Why did we just use it on $\mathbb N$ without justification?
The axiom of completeness states that if $S\subset\Bbb R$ and $S\ne\emptyset$, then if $S$ has an upper bound, then $\sup S$, and if $S$ has a slower bound, then $\inf S$ exists. And $\Bbb N$ is a non-empty subset of $\Bbb R$. So, yes, you can use the axiom here.