Let M be a martingale and $\tau$ be a stopping time with $P(\tau < \infty)=1$
Show $M_{\tau \land n}$ (i) and $M_{\tau}$ (ii) are integrable:
i) Since we have no supremum stating that $\sup \limits _{n}E(\vert M_{ n}\vert)<\infty$, I can only estimate $E(\vert M_{\tau \land n}\vert)$ from above by $\sup\limits_{i=\tau , n }E(\vert M_{i}\vert)<\infty$ but this is contingent on the fact that I prove that $E(\vert M_{\tau}\vert)<\infty$ for which I am yet to find an assured solution.
Answer for first part: $|M_{\tau \wedge n}| \leq |M_1|+|M_2|+\cdots +|M_n|$ so $E|M_{\tau \wedge n}| \leq E|M_1|+E|M_2|+\cdots +E|M_n| <\infty$.
(There is no reason why $E|M_{\tau }|< \infty$. A counterexample can be found in many books dealing with stopping times).