I was reading in Varadhan's book: probability theory and where I found the following exercise $5.14$. I also came across a similar question: How to prove $\mathcal F_T \subseteq \sigma(\bigcup_n \mathcal F_{T_n})$?
Can we claim that $\Omega\subset\bigcup_{n \in \mathbb{N}}\{\tau_n=\tau\}$? Why ? In case the claim was wrong how can we prove that $\mathcal{F}_{\tau} \subset\left(\bigcup_{n \in \mathbb{N}}\mathcal{F}_{\tau_n}\right)$?
To answer your question: No. Take e.g. the degenerate stopping times $\tau_n=1-1/n$ and $\tau=1$. Then, clearly $\{\tau_n=\tau\}=\emptyset$ for all $n\in\mathbb{N}$ and so their union is again empty.
I'm curious as to why you would need this in order to solve the posted exercise?
Edit: The stated inclusion is correct (with equality, even) if the stopping times take values in $\mathbb{N}$ and $\tau(\omega)<\infty$ for all $\omega\in\Omega$. This is due to the fact that if $\{a_n\}$ is an increasing sequence in $\mathbb{N}$ converging to some $a\in\mathbb{N}$, there must be some $N\in\mathbb{N}$ such that $|a_n-a|<1$ and hence $a_n=a$ for all $n>N$.