The question is: Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. Prove that for every $x > 0$ $$P\{\sup_{t\geq 0} M_t > x \,| \,F_0\} = 1\wedge \frac{M_0}{x}$$ almost surely.
From Doob's martingale inequality we may conclude that $P\{\sup_{t\geq 0} M_t > x \,| \,F_0\} \leq 1\wedge \frac{M_0}{x}$, but right now I'm stuck on showing that $$P\{\sup_{t\geq 0} M_t > x \,| \,F_0\} \geq 1\wedge \frac{M_0}{x}.$$ On this part I'm completely clueless right now. Any tips?
P.S. We were given the tip "stop the martingale when it gets above $x$", but I don't see what we can do with such a stopped martingale.
