It is a bit confusing the way he writes it, but what he actually shows in Theorem 3 (in the case $a=0$), is that if $\sum a_n x^n$ converges, then $\sum a_n y^n$ absolutely converges for any $|y|<|x|$. The invocation of $ACT$ is confusing since it speaks about a notion (radius of convergence) whose existence is proved in Theorem 1. However, in the proof of Theorem 3, $R$ is used only to take an $|x|<R$, so that we know $\sum a_n x^n$ converges. What he should have said is "from the proof of Theorem 3, etc...".
In the proof of Theorem 3 (in the book) he picks an arbitrary $|x|<R$ for which he has to show absolute convergence. Then he says there exists $X$ such that $|x|<|X|<R$. For this $X$ you have convergence, and from this convergence alone (no $R$ needed after this point) he deduces absolute convergence for $x$. Notice that at no point he assumes or uses that the series converges for $x$. He only uses radius of convergence to say that the series converges for $X$. In other words he proves the following: If $\sum a_n x^n$ is convergent, than for any $|y|<|x|$, $\sum a_n y^n$ is absolutely convergent.
For Theorem 1, IF a) and b) do not hold, there must be an $X$ for which the series diverges. He concludes that for any $|x|>|X|$, the series still diverges. Indeed, if it converges for some $|x|>|X|$, it would absolute converge for ANY $|y|<|x|$ (by the previous argument), in particular for $X$. Contradiction.