show $\max$ and $\min$ are stopping times.

Question

I need your opinion about my proof.

Let $\tau$ and $\sigma$ two stopping time on $( \Omega , \mathscr{F} , (\mathscr{F}_n)_n , \mathbb{P} )$. Let $\mu_1: \Omega \rightarrow \mathbb{N} \cup \left\{ + \infty \right\} $ defined by

$$ \forall \omega \in \Omega \mbox{, } \mu_1( \omega ) = \max \left( \tau(\omega),\sigma(\omega) \right) $$

I have to show that $\mu_1$ is a stopping time.

We write :

$$ \left\{ \mu_1= n \right\} = \left\{ \tau \geq \sigma , \tau = n \right\} \cup \left\{ \tau \leq \sigma , \sigma = n \right\} = \left( \left\{ \tau \geq \sigma \right\} \cap \left\{ \tau = n \right\} \right) \cup \left( \left\{ \tau \leq \sigma \right\} \cap \left\{ \sigma = n \right\} \right) $$

Now :

1. $\left\{ \tau \geq \sigma \right\} = \left\{ \tau \leq \sigma - 1 \right\}^c \in \mathscr{F}_{n-1} \subset \mathscr{F}_{n} $
2. $\left\{ \tau = n \right\} \in \mathscr{F}_{n} $
3. $\left\{ \tau \leq \sigma \right\} \in \mathscr{F}_{n} $
4. $\left\{ \sigma = n \right\} \in \mathscr{F}_{n}$

Then $\left\{ \mu_1= n \right\} \in \mathscr{F}_{n} $

It is correct? Thank you