# Is it necessary to have $A$ unital in this exercise?

So today I obtained Gerard Murphy's $C^*$-Algebras and Operator Theory, and I have been going through the exercises in the earlier chapters to help bring me up to speed. In chapter $1$, exercise $13$ is stated as follows:

Suppose that $d$ is a bounded derivation on a unital Banach algebra $A$ and $\lambda\in\mathbb{C}\setminus\{0\}$ such that $da=\lambda a$. Show that $a$ is nilpotent, that is, that $a^n=0$ for some positive integer $n$ (use the boundedness of $\sigma(d)$).

For completeness, a derivation on an algebra $A$ is a linear map $d:A\to A$ such that $d(ab)=d(a)b+ad(b)$ for all $a,b\in A$.

I came up with the following solution:

By induction, we have $d(a^n)=n\lambda a^n$ for all positive integers $n$. If $a$ is not nilpotent, then $\{n\lambda:n\in\mathbb{N}\}\subset\sigma(d)$. But $r(d)\leq\|d\|$, so $n|\lambda|\leq\|d\|$ for all $n$, a contradiction. Thus $a$ is nilpotent. (Here, $r(d)$ denotes the spectral radius of $d$).

However, my proof does not explicitly use the assumption that $A$ is unital. Am I missing something? Did I make a mistake? Or is the assumption superfluous?