Consider the following discrete time Markov process with the unit interval as state space. If the process is at $p\;(0<p<1)$ at the present, it will jump to $\alpha+\beta p$ with probability $p$ and to $\beta p$ with probability $1-p$ after the next trial, where $\alpha,\beta>0$ and $\alpha+\beta=1$. In symbols, the process is defined by the transformation law $$X_{n+1}= \begin{cases} \alpha+\beta X_n,\quad&\text{with probability } X_n,\\ \beta X_n, &\text{with probability } 1-X_n \end{cases} $$ Show that this process is a martingale.
In order to show that $\{X_n\}$ is a martingale, we have to show that $E[X_{n+1}|\mathcal{F}_n]=X_n$.
\begin{align} E[X_{n+1}|\mathcal{F}_n]&=X_n(\alpha+\beta X_n)+(1-X_n)\beta X_n\\ &=X_n(\alpha+\beta X_n+\beta-\beta X_n)\\ &=X_n(\alpha+\beta)\\ &=X_n\quad\text{because }\alpha+\beta=1 \end{align} This is my attempt at the problem. I got the desired answer but I am wondering if I did this correctly because I did not use the condition on $\mathcal{F}_n$. Could I get some verification on this? Thank you.