H.A. Priestley in "Introduction to Complex Analysis" page 73 gives two proofs of the "Radius of Convergence lemma". The first seems to turn on the difference between Convergence and Absolute Convergence and though it seems to be almost arguing in circles I can just about see how it works. The second proof, though, which is by contradiction, I do not see at all.
For the contradiction, Priestley puts forward some $z$ with $\vert z|>R$ for which $\sum c_n z^n$ converges. Then, as he has shown earlier, there exists $M$ such that $\vert c_n z^n| \leq M$ for all $n$.
Pick $w$ such that $R<\vert w| < \vert z|$. Then $\vert c_n w^n| = \vert c_n z^n|\vert \frac {w^n}{z^n}| \leq M \vert \frac {w}{z}|^n$.
The geometric series $\sum \vert \frac {w}{z}|^n$ converges, because $\vert \frac {w}{z}|< 1$. Hence, by the Comparison test, $\sum\vert c_n w^n|$ converges. This contradicts the definition of $R$.
My problem is that we've already contradicted the definition of $R$ in defining $w$. For a proof by contradiction, wouldn't one need, rather, something that contradicted the original (contrary to fact) assumption? Or is there something about the Comparison test that I have failed to understand?
Daniel Fischer's comment seems to be the answer.
I think you are confusing the convergence of $\sum c_n z^n$ with the convergence of $\sum |c_n z^n|$. These are different things. The "Radius of Convergence Lemma" is about the convergence of $\sum c_n z^n$. But the number $R$ is defined in terms of the convergence of $\sum |c_n z^n|$.
Let's look at the proof of (2). When the proof begins, we assume for contradiction that the statement of (2) is false, meaning that there exists $z \in \mathbb{C}$ for which $|z| > R$ and $\sum c_n z^n$ converges. This does not contradict the definition of $R$, which pertains to the convergence of $\sum |c_n z^n|$.
We then choose a number $w \in \mathbb{C}$ for which $R < |w| < |z|$. There is again no contradiction here: there are plenty of complex numbers $w \in \mathbb{C}$ for which $R < |w| < |z|$. Nothing about the convergence of $\sum c_n w^n$ or $\sum |c_n w^n|$ is assumed: the complex number $w \in \mathbb{C}$ whose magnitude is between $R$ and $|z|$ is totally arbitrary.
Anyway, one then goes on to show that, in fact, $\sum |c_n w^n|$ converges. The convergence of this series --- notice the absolute value bars --- does contradict the definition of $R$, which completes the proof.