I have a doubt in proof of Lemma $4.7$ of this paper.

**Lemma:** Let $A$ be a closed operator on a complex Banach space $E$ and assume that $0$ is an eigenvalue of $A$ and a pole of the resolvent $R(\cdot, A).$ Denote by $P$ the corresponding spectral projection.

$(i)$ If $\lambda > 0$ is contained in $\rho(A)$ and if the operator family $([\lambda R(\lambda, A)]^n)_{n \in \mathbb N}$ is bounded then $0$ is a simple pole of the resolvent.

$(ii)$ Suppose in addition that $0=s(A).$ If $\lambda >0$ and $s(A)=0$ is a simple pole of the resolvent, then $$\lim_{n \to \infty} [\lambda R(\lambda, A)]^n = P.$$

In the proof of $(ii),$ the author first shows that there exists $c \in (0,1)$ such that $|\mu| \leq c$ for all $\mu \in \sigma(\lambda R(\lambda,A)) \setminus \{1\}.$ He then says, in particular $$r(\lambda R(\lambda,A)(I-P))\leq c<1.$$

I don't understand how the last statement follows in particular. Is there any reason why $$\sigma(\lambda R(\lambda,A)(I-P))\subseteq \sigma(\lambda R(\lambda,A))?$$