I have a question regarding Functional Analysis:

Assume that $X$ is a Banach space and that $P \in B(X)$ is a projection with $ran P \neq {[0]}$.

Then I have easily shown that $\lambda = 0$ and $\lambda = 1$ are eigenvalues of $P$.

I have troubles proving that if $\lambda \notin\{0,1\}$, then it follows that we have $(P-\lambda)^{-1} = \frac{1}{\lambda(1 - \lambda)}P - \frac{1}{\lambda}I$.

Any help would be grateful.