Convergence of a martingale

Question

Let $X_n$ takes its values in [0,1] and $p$ is a fixed number in $[0,1]$ Now if $ X_{n+1} = 1-p+pX_n $ with probability $X_n$ and $X_{n+1} = pX_n$ with probability $1-X_n$ . I know that $X_n$ is a martingale but why it converges almost surely? with which one of the theorems?

Answer 1

Since $X_n$ takes its values in $[0,1]$, we have in particular

$$\sup_{n \in \mathbb{N}} \|X_n\|_1=1<\infty$$

Hence $X_n \to X_{\infty}$ almost surely for some random variable $X_\infty$ ($L^1$-bounded martingales are a.s. convergent).