A question about Banach algebras: showing that $\operatorname{Sp}a \subset D_o \cup D_1$

Question

Maybe this problem be easy for a person that have study in Banach Algebra; please give me a hint.

Let $e=0$ or $1$, and $a$ be an arbitrary element in a Banach algebra $A$. Let $D_o$ and $D_1$ be the disks in the complex plane of the same radius $\|a\|$ centred at $0$ and $1$, respectively. Then $\operatorname{Sp}a \subset D_o \cup D_1$.

Answer 1

This is Theorem 3.2.3 in "Fundamentals of the Theory of Operator Algebras" by Kadison and Ringrose. Their proof is instructive and I would suggest trying to look at it: