TITLE: Sum of two stopping times is a stopping time?
QUESTION [9 upvotes]: Let $\sigma$ and $\tau$ be two stopping times in $\mathscr{F}_t$ and let this filtration satisfy all the usual conditions. 
Question: Is $\sigma + \tau$ a stopping time?
Attempt at a solution: 
I need to demonstrate that $\{ \sigma + \tau \leq t\}\in \mathscr{F}_t$, or that $\{\sigma \leq t - \tau \} \in \mathscr{F}_t$. 
Since $\sigma$ is a stopping time we have that $\{\sigma \leq t - \tau\} \in \mathscr{F}_{t - \tau}$, where $t - \tau \in [0,t]$. 
Since $t > t - \tau$, we have that $\mathscr{F}_{t-\tau} \subseteq \mathscr{F}_t$ by the definition of $\mathscr{F}$. 
This implies that $\{\sigma \leq t - \tau\} \in \mathscr{F}_t$, and that $\sigma + \tau$ is a stopping time.

Is my attempt correct?

REPLY [2 votes]: There is a brief proof in the book ``S. W. He et al., Semimartingale Theory and Stochastic Calculus, CRC Press Inc, 1992.''(p.84, Th.3.7.(3)). 
The proof is based on following fact(Th.3.7.(1)): If $S$ is a stopping time, r.v. $ T\in\mathscr{F}_S $ and $ T\ge S $, then $ T $ is a stopping time.(Proof of this fact: For each $ t\ge 0 $, $ [T\le t]\in\mathscr{F}_S$, and by the definition of $\mathscr{F}_S $ we have 
$ [T\le t]=[T\le t][S\le t]\in\mathscr{F}_t $, hence $ T $ is a stopping time.)  
$ S+T $ is a stopping time: Since $ S+T\ge S \lor T $ and $ S+T\in \mathscr{F}_{S\lor T} $($ S\in \mathscr{F}_S\subset \mathscr{F}_{S\lor T}$),
then $ S+T $ is a stopping time.