Suppose G is open and D[a,r]⊂G. Show that there is R>r so that D[a,r]⊂D[a,R]⊂G.
Hint. If G=C just take R=r+1. Otherwise choose some w∈C∖G, let M=∣w−a∣, and let K=D[a,M]∖G. Show that K is nonempty, closed and bounded, and apply Theorem A.0.1 to find a point z0∈K that minimizes f(z)=∣z−a∣ on K. Show that R=∣z0−a∣ works.