Construct a sequence $\{X_n\}$ of random variables with finite mean such that $E(X_{n+1}|X_n)=X_n$ for all $n$ but $\{X_n\}$ is not a martingale.
Construct a sequence $\{X_n\}$ of random variables with finite mean such that $E(X_{n+1}|X_n)=X_n$ for all $n$ but $\{X_n\}$ is not a martingale.