Let $(\Omega,\mathcal F,\mathbb P)$ be a complete probability space with filtration $(\mathcal F_t)_{0\leqslant t\leqslant \infty}$ such that $\mathcal F_0$ contains all the $\mathbb P$-null sets of $\mathcal F$ and the filtration is right-continuous, i.e. $\mathcal F_t = \bigcap_{u>t}\mathcal F_u$. It is claimed that
The event $\{T<t\}\in\mathcal F_t$ if and only if $T$ is a stopping time.
The proof is as follows: Since $\{T\leqslant t\}=\bigcap_{t+\varepsilon>u>t}\{T<u\}$ for any $\varepsilon>0$, we have $\{T\leqslant T\}\in\bigcap_{u>t} \mathcal F_u = \mathcal F_t$, so $T$ is a stopping time. (This part I am fine with.)
My question is about the converse. It is claimed that $\{T<t\}=\bigcup_{t>\varepsilon>0}\{T\leqslant t-\varepsilon\}$, and $\{T\leqslant t-\varepsilon\}\in\mathcal F_{t-\varepsilon}$, hence also in $\mathcal F_t$.
But $\bigcup_{t>\varepsilon>0}\{T\leqslant t-\varepsilon\}$ is an uncountable union of events. Shouldn't we instead consider a sequence of rational numbers $r_n>0$ with $\lim_{n\to\infty} r_n=0$ and $\bigcup_{t>r_n>0}\{T\leqslant t-r_n\}$?
It is enough to consider the values $\epsilon =\frac 1 n$: $T<t$ iff there exists $n$ such that $T \leq t-\frac 1n$.