Here we consider random elements of $D_\infty$, i.e. cadlag functions on $[0,\infty)$ with the Skorohod metric. I cannot figure out how to get the first inequality in the last paragraph. I can see the intuitive idea, but I cannot rigorously understand it. Specifically, we have that if (16.25) holds but (16.27) does not, then either $P[\theta \in M_1^c] \ge 1/4$ or $P[\theta \in M_2^c] \ge 1/4$. Let us stick to the first. Then we look at the event $[|X_{\tau_2}^n - X_{\tau_1}^n| \ge 2\epsilon, \tau_2 - \tau_1 \le \delta]$. How can we show that $P[\theta \in M_1^c] \ge 1/4$ gives us the event $[P[|X_{\tau_1+\theta}^n - X_{\tau_1}^n| \ge \epsilon || \mathscr{F}^n] \ge 1/4]$?
