When proving that a stopping time $\tau$ is $\mathcal{F}_\tau$-measurable it suffices to show that $\{ \tau \leq t\} \in \mathcal{F}_\tau$ in my opinion which is not very difficult. My question is, why showing $\{ \tau \leq t\} \in \mathcal{F}_\tau$ is enough.
Is it because of \begin{equation} \{ \tau = t \} = \{\tau \leq t\} \bigcap_{n \geq 1} \{\tau > t + 1/n\} \end{equation} and $ \{ \tau < t \} = \{ \tau \leq t \} - \{\tau = t\}$? Is this reasoning ok? Or is there a very obvious and shorter argument? Thanks.
Your reasoning is not correct. By definition $\{\tau \leq t\}\in \mathcal F_{\tau}$ means $\{\tau \leq t\} \cap \{\tau \leq s\}\in \mathcal F_s$ for each $s$. For this note that $\{\tau \leq t\} \cap \{\tau \leq s\}=\{\tau \leq t\wedge s\} \in \mathcal F_{t\wedge s} \subset \mathcal F_s$.