Suppose $\{X_n\}_{n = 1}^\infty$ is a supermartingale (a sequence of random variables, such that $\forall n \in \mathbb{N} P(E(X_{n + 1}|X_1, … , X_n) \leq X_{n + 1}) = 1$). Is it always true, that $P(\sup_{1 \leq k \leq n} X_k \geq a) \leq \frac{E(\max\{0, X_n\}) - E(X_n) + EX_1)}{a}$?
What have I tried to solve this problem:
According to Markov inequality, $P(\sup_{1 \leq k \leq n} X_k \geq a) \leq \frac{E(\sup_{1 \leq k \leq n} X_k)}{a}$, and thus it would have been sufficient to prove, that $E(\sup_{1 \leq k \leq n} X_k) + E(X_n) \leq E(X_1) + E(\max{0, X_n})$.
However, this method fails, as in the particular case, when $\forall n \in \mathbb{N} P(X_n \geq 0) = 1$, the aforementioned statement reduces to $E(\sup_{1 \leq k \leq n} X_k) \leq E(X_1)$, which is not generally true.
And here I am stuck, not knowing what to try next.