Let $X_t$ be a continuous time stochastic process. Define the first hitting time of $X$ as $\tau = \inf~\{ t \leq t': \lvert X_t \rvert \geq \lambda \}$ for some $\lambda$, $t'$
It is a known fact that $\{ \max_{t \leq t'} \lvert X_t \rvert \geq \ \lambda \} = \{ \tau \leq t' \}$. My Professor claimed in addition that $\{ \tau \leq t' \} \subseteq \{ \lvert X_{\tau \land t' } \rvert \geq \ \lambda \} \} $. However, I don't see why the $\subseteq$ is not just an $=$ . My reasoning is like this: at $|X_{\tau}|$ the value of the process is exactly $\lambda$. However, $|X_{\tau \land t'}|$ is just $|X_{\tau}|$ when $\tau < t'$, and in the case when $ \tau = t'$ we have that $|X_{\tau}| = |X_{t'}| = \lambda$. Where am I going wrong with this proof?