Q:

proof of a lemma on stopping times

Hi I got some problem with the proof of this lemma which is left as excercise 
Let $(F_t)$ be a filtration and S,T be stopping times, then show $
\{S=T\},\{S≤T\}，\{S<T\} \in F_S\bigcap F_T$.
Could any one show to how to do it ?  thanks in advance. 

A:

By symmetry, you just have to check that these events all belong to $\mathcal{F}_T$. 
Let's do it for $\{T = S\}$ : by definition of $\mathcal{F}_T$, you have to prove that $\{T = S \} \cap \{T \le n \}$ belongs to $\mathcal{F}_n$ for all $n \ge 0$. 
But $\{T = S \} \cap \{T \le n \} = \bigcup_{k \le n} \{T =k \} \cap \{S = k\}$. As both $\{T = k\}$ and $\{S  =k \}$ belong to $\mathcal{F}_k \subset \mathcal{F}_n$, we have the result.