Quasi-left-continuity and sequences of stopping times

Let $X$ be a bounded adapted right-continuous process, let $\varepsilon >0$, let $U_{0}=0$, a.s., and define $U_{i}$ by (16.3) for $i \geq 1$. Show each $U_{i}$ is a stopping time.

Let $\left(X_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a real - valued process whose paths $t \mapsto X_{t}(\omega )$ are (a.s.) right continuous and have finite left - hand limits (this property is often called "càdlàg"). Assume, in addition, that $X$ is quasi left continuous, i.e. for every sequence of stopping times $\left(\tau_{n}\right)_{n \geqslant 1}$ which increase to a stopping time $\tau$ we have $\lim_{n \rightarrow \infty } X_{\tau_{n}} = X_{\tau }$ a.s. on the set $\left\{ \tau < \infty \right\}$. Show that the assertion of Lemma 5.8 remains valid for such processes if the filtration contains all subsets of measurable null sets.