I am having difficulty to prove some of the proof of Theorem 5.7.1. from the book, "Chaos, Fractals, and Noise" by A. Lasota. I think it is a simple problem, but somehow am not able to verify it. I hope to get some help.
Part of the proof that I am having difficulty to understand is as follows:
Problem: Let's assume the following inequality holds: \begin{equation} \int_G V(x) Pf(x) dx \leq \alpha \int_0^\infty V(x) f(x) dx + \beta,\, 0\leq \alpha <1,\, \beta \geq 0. \end{equation}
Also, let \begin{align} E_n(V|f) := \int_G V(x) P^n f(x) dx \end{align} where $\{P^n\}$ is a sequence of Markov operator. For your reference, $P$ is a linear operator. Using the above inequality, \begin{align} E_n(V|f) \leq \alpha E_{n-1} (V|f) + \beta \end{align} By induction, the following holds: \begin{align} E_n(V|f) \leq \frac{\beta}{1-\alpha} + \alpha^n E_0(V|f). \end{align} What I don't understand is how the constant part $\frac{\beta}{1-\alpha}$ is derived. I tried the followings.
My try: \begin{align} E_n(V|f) &\leq \alpha E_{n-1}(V|f) + \beta\\ &\leq \alpha (\alpha E_{n-2}(V|f) + \beta) + \beta\\ &\leq \alpha (\alpha (\alpha E_{n-3}(V|f) + \beta) + \beta) + \beta\\ &\vdots\\ &\leq \alpha^n E_0(V|f) + \alpha^{n-1} \beta + \alpha^{n-2} \beta + \ldots + \alpha \beta + \beta \end{align} By sum of geometric series, \begin{align} \alpha^{n-1} + \alpha^{n-2} + \ldots + \alpha + 1 = \sum_{k=0}^{n-1} \alpha^k = \frac{1-\alpha^n}{1-\alpha} \end{align} Hence, it becomes \begin{align} E_n(V|f) &\leq \alpha^n E_0(V|f) + \beta \frac{1-\alpha^n}{1-\alpha}\\ &\leq \alpha^n E_0(V|f) + \beta \frac{1}{1-\alpha}. \end{align} I am not sure if it is the right derivation.
I greatly appreciate your help in advance.