I am looking at a proof which shows, if we have a power series $\sum_{n=0}^\infty a_nx^n$, there exists an R > 0 such that the series converges absolutely for every x with |x| < R, and diverges for every x with |x| > R.
I am confused by the text below where is says there exists a $\rho$ in G s.t. $|x|<\rho$ since wouldn't this contradict the fact R is the least upper bound?
You're confusing "a specific x" with "all x that satisfy the condition".
R is a least upper bound for G because it is the smallest number that works for all of the |x| in G.
For any specific |x| in G, there must be a p in G with
|x|<p. If there weren't, then |x| would be the upper bound of G, and |x| < R, meaning R wouldn't be the least upper bound.