We work with respect to a filtered probability space $(\Omega,\mathcal F,\{\mathcal{F_t\}_{t\ge 0},P})$
Let $Y$ be a cadlag adapted process. Fix $\epsilon \in \mathbb R_0^+$. Let $\tau$ be the following stopping time
$$\tau=\inf\left\{t\in\mathbb R^+ \mid \vert Y(t)\vert\ge \epsilon\right\}$$
Let $\tau_n$ be the smallest rational number bigger than $\tau$ with denominator $n$, that is
$$\tau_n=\min\{m/n \mid m \in \mathbb N, m/n\ge\tau\}$$
Then we have $$\tau \le \tau_n \le \tau + 1/n$$
which implies that $\tau_n \downarrow \tau$.
Fix a $t \in \mathbb R^+$. Define the processes $\xi_n(s)=1_{\{s\le\tau_n\wedge t\}}$
Are those processes elementary ?
An elementary process is of the form
\begin{equation} \label{eq:1} \xi(t) = Z_01_{\{t=0\}}+\sum_{k=1}^n Z_k1_{\{s_k<t\le t_k\}} \end{equation} for ${n\ge 0}$, times $0 \le s_1 < t_1 \le s_2 < t_2 \le \dots \le s_n < t_n \,$, ${\mathcal{F}_0}$-measurable random variable ${Z_0}$ and ${\mathcal{F}_{s_k}}$-measurable random variables ${Z_k}$.
$\xi_n(s)= 1_{s \leq t} 1_{s \leq \tau_n}$ and $1_{s \leq \tau_n}= \sum_k 1_{\tau_n\geq k/n} 1_{k/n < s \leq (k+1)/n} $ so yes, $\xi_n$ are elementary processes.