# Convergence of a martingale

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?

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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).
Concerning the distribution you can take a look at this question (choose $\alpha:=1-p$, $\beta:=p$). –  saz Nov 29 '12 at 6:46