Below is an exercise and hint from Rogers and Williams' Diffusions, Markov Processes, and Martingales Volume 2.
Here, we are trying to show that if $H$ is a previsible process in $l(l(\mathscr{H}))$ then $H \in l(\mathscr{H})$. To complete the proof from the hint, we need to show that $H(0,T_n] \in \mathscr{H}$, that is $H(0,T_n] \in \mathscr{H}$.
My questions are: How do we show that $T_n$ is a stopping time increasing to $\infty$ a.s., and how do we show that $H(0,T_n] \in \mathscr{H}$? I can't figure out how to use the probability inequality to prove this. I would greatly appreciate any help.
$\{T_n< t\} =\cup_{r=n}^\infty\{V_{r,k(r)}<t\}$, which shows that $T_n$ is a stopping time, provided the filtration is right continuous.
It should be clear the $\{T_n\}$ is an increasing sequence. By Borel-Cantelli, $V_{r,k(r)}>U_r-2^{-r}$ for all large $r$, a.s. Therefore $T_n\ge U_n-2^{-n}$ for all large $n$, a.s., and so $\lim_n T_n=\infty$ a.s.
Now $H(0,V_{r,k(r)}]\in\mathscr H$ for each $r\ge n$, and from the definition of $T_n$ you have $(0,T_n] =\cap_{r\ge n}(0,V_{r,k(r)}]\subset (0,V_{n,k(n)}]$. Consequently, $H(0,T_n]=H\cdot1_{(0,T_n]}=(H\cdot1_{(0,V_{n,k(n)}]})\cdot1_{(0,T_n]}$, and this latter process is in $\mathscr H$ because $H\cdot 1_{(0,V_{n,k(n)}]}\in\mathscr H$ and $T_n$ is a stopping time.