Let $X = ({X_n})_{n \ge 1}$ be a/an $(\{\mathscr{F_n}\}_{n \ge 1}, \mathbb{P})$-supermartingale in the filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}_{n \ge 1}, \mathbb{P})$.
Prove $X_T$ is integrable if $\exists K \ge 0$ s.t. $$|X_n - X_{n-1}| \le K \ \forall n \ge 1$$
and $T$ is a stopping time w/ finite expectation.
So far all I was able to show is that $X_{T \wedge n}$ is integrable and $E[X_T] \le E[X_0]$.
Hints pls?
$$|X_T| = |\lim_{n \to \infty} X_{T \wedge n}| = \lim_{n \to \infty} | X_{T \wedge n}|$$
Consider $X_{T \wedge n}$:
$$X_{T \wedge n} = \sum_{k=1}^{T \wedge n} (X_k - X_{k-1}) + X_0$$
$$\to |X_{T \wedge n}| = |\sum_{k=1}^{T \wedge n} (X_k - X_{k-1}) + X_0|$$
$$ \le |\sum_{k=1}^{T \wedge n} (X_k - X_{k-1})| + |X_0|$$
$$ \le \sum_{k=1}^{T \wedge n} |X_k - X_{k-1}| + |X_0|$$
$$ \le \sum_{k=1}^{T \wedge n} K + |X_0|$$
$$ = (T \wedge n) K + |X_0|$$
Thus, we have
$$|X_T| \le \lim [(T \wedge n) K + |X_0|]$$
$$\le [\lim (T \wedge n)] K + |X_0| \le TK + |X_0|$$
Hence,
$$E[|X_T|] \le E[TK + |X_0|]$$
$$\le K\underbrace{E[T]}_{ < \infty} + \underbrace{E[|X_0|]}_{< \infty \ \because \ \text{X is a supermart}} < \infty$$