I'm having a hard time figuring out why the infimum of a sequence of stopping times is not necessarily a stopping time itself. Indeed, the justification my book gives me is that:
Given $(\mathcal F_t)_{t \in (0,\infty)}$ then we can write $$\left\{\inf{T_n}\leq t \right\} = \bigcap_{j\geq k} \left\{\inf{T_n}<t+\dfrac{1}{j}\right\} = \bigcap_{j\geq k}\bigcup_{n\geq 1} \left\{T_n<t+\dfrac{1}{j} \right\}$$
At this point it's pretty clear that writing $\{ \text{inf} \;T_n\leq t \} $ in this way we cannot say that this event belongs to $\mathcal F_t$. Anyways, my concern is that one can also write: $$ \{ \inf{T_n}\le t \} = \bigcup_{n\ge 1} \{T_n\le t\} $$ but my statement would bring to the opposite conclusion, so I cannot figure why that writing is false!
Your way of writing the event is wrong. Note that $\inf \limits_n \frac{1}{n} = 0$, while $\frac{1}{n} \le 0$ never holds.