Let's take $X_1 \sim U[0,1]$ and if $X_{n-1}=a$ then $X_n \sim U[0,a]$. I want t prove that $X_n$ is supermartingale.
So what I want to do prove is :
$$E[X_{n+1}\mid \mathbb{F}_n]\le X_n$$
And I have several problems with it. Let's just think a little about next expectations :
$E[X_1]=\frac12$
$E[X_2]=\frac{a_1}{2}$
$E[X_3]=\frac{a_2}{2}$
and so on. So how can I translate it now to language of conditional expectation with filtration ? Is it really submartingale ? I found it very unintuitive.