Let $T_1, T_2, T_3, \, \ldots \overset{d}{=} \text{Exponential}(\theta)$ be an independent sequence of exponential variables that defines $X$ and let $\omega \in \Omega$ be an outcome. Then we have the inclusion
\begin{equation}
\left\{ M \in \mathbb{N} : \sum_{m = 1}^M T_m (\omega) \leq s \right\}
\subseteq \left\{ M \in \mathbb{N} : \sum_{m = 1}^M T_m (\omega) \leq t \right\}
\end{equation}
Using result
R1110: Isotonicity of maximum, therefore
\begin{equation}
X_s (\omega)
= \max \left\{ M \in \mathbb{N} : \sum_{m = 1}^M T_m (\omega) \leq s \right\}
\leq \max \left\{ M \in \mathbb{N} : \sum_{m = 1}^M T_m (\omega) \leq t \right\}
= X_t (\omega)
\end{equation}
Since $\omega \in \Omega$ was arbitrary, the result follows. $\square$
