Let $\tau$ a stopping time.
Define $$\tau_n = \frac{k+1}{2^n}, \text{ if } \frac{k}{2^n}<\tau \leq \frac{k+1}{2^n}$$.
I already checked it's a stopping time, but the teacher wrote in the notes
$$ \lim_{n \rightarrow \infty} \tau_n = \tau$$
I really can't understand why it holds. As $n$ becomes larger and larger $\tau$, I would say that $\tau_n \rightarrow 0$
The trick is to note that for any $n$, $\tau_n \ge \tau$ and $|\tau_n -\tau| \le 2^{-n}$. By definition these hold and in the limit, we have $\tau_n \to \tau$.
On another note, i am assuming that all convergences are almost sure.