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I need to prove the following statement.

Let $X$ be a non negative martingale such that $X_n\rightarrow 0$ a.s. when $n\rightarrow \infty$. Define $X^*=supX_n$. Prove that for all $x>0$

$$P[X^* \geq x | \mathcal{F}_0]= 1 \wedge X_0 / x$$

I think I've got the easy case

  • if $x\leq X_0$

Then necessarily $x\leq X^*$ for the sup property. Then it follows that for $1\leq X_0 /x$ we have that $P[X^* \geq x | \mathcal{F}_0]= 1$. But I can't figure out the other case.