stopping times (definition)

Question

I have the following question. If we want to verify that a random variable $T$ is a stopping time, according to the definition of a stopping time we have to show that $[T\le n]\in F_{n} $ for each $n\in\mathbb{N}$. I noticed that is enough to show only $[T= n]\in F_{n}$ for each $n\in\mathbb{N}$. Why is it enought?

Answer 1

It's enough because, $[T\leq n] = \cup_{k=1}^{n}[T=k]$ where $[T=k]\in\mathcal{F}_n\subset \mathcal{F}_n$ for all $1\leq k\leq n.$

Answer 2

It is assumed that $\mathcal F_n$'s are increasing. $[T=n]=[T=1] \cup [T=2]\cup...\cup[T=n]$ and $[T=i] \in F_i \subseteq F_n$ for each $i \in \{1,2...,n\}$. Hence $[T=n] \in F_n$