Properties of Stoppign times

Question

Let $S$ and $T$ be stopping times with $S \leq T$. Then $\mathcal{F}_S \subseteq \mathcal{F}_T$. Note that $\{T \leq n\} \subseteq \{S \leq n\}$

My question is why the last inequality holds? $\{T \leq n\} \subseteq \{S \leq n\}$

Answer 1

Statement $S\leq T$ tells us actually that: $$S(\omega)\leq T(\omega)\text{ for every }\omega\in\Omega$$

$\omega_0\in\{T\leq n\}$ is the same statement as $T(\omega_0)\leq n$ and combining this with $S(\omega_0)\leq T(\omega_0)$ we conclude that also $S(\omega_0)\leq n$, or equivalently $\omega_0\in\{S\leq n\}$.

Proved is now that: $$\{T\leq n\}\subseteq\{S\leq n\}$$