Let $S_1, S_2, \dots, S_n$ be a supermartingale, such that $$|S_i| \le C$$ almost surely, for some positive constant $C$ and all $1\le i \le n$. My question is:
Does there always exist a martingale $M_1, M_2, \dots, M_n,$ such that $$S_1=M_1$$ $$S_i \le M_i$$ and $$|M_i|\le C$$ for all $1\le i \le n$?
In other words: if we have a bounded supermartingale, can we always "improve" it while not validating the given bound $C$?
My intuition is that it should be true. I think that it is (more or less) obvious for $(S_i)_{i=1}^{n}$ with finite support.
Is it also true without this constraint? What would be a short, yet formally sound argument?