Calculate $\lim_{n \rightarrow \infty} E[X_n]$

Question

Let's consider the following sequence : $X_1 \sim U[0,1]$. And if $X_1=a$ then $X_2 \sim U[0,a]$. Now if $X_2=b$ then $X_3 \sim U[0,b]$ and so on so on. I want to calculate $$\lim_{n \rightarrow \infty}E[X_n]$$

Hint Use, properties of supermartingales.

My work so far

Proving martingale is very easy :

$$E[X_{n+1}|\mathbb{F}_t]=\frac12 X_n \le X_n$$

Now how to use it ? Let's see

$$E[X_{n+1}|\mathbb{F}_t] \le X_n \Leftrightarrow E\bigl[E[X_{n+1}|\mathbb{F}_t]\bigr]\le E[X_n] \Leftrightarrow X_{n+1} \le E[X_n]$$

But now I stacked. Could you please give me a hand ?

Answer 1

You've already shown $$E[X_{n+1}] = E[E[X_{n+1} \mid X_n]] = \frac{1}{2} E[X_n].$$ This implies $E[X_{n+1}] = \frac{1}{2^n} E[X_1] $.