For $n,m\in\mathbb{N}_0$, $n<m$, and $E\in\mathcal{F}_n$, how to show that $n1_E + m1_{\Omega \setminus E}$ is a stopping time?
Should I break it into cases, $\omega\in E$ and $\omega\in \Omega\setminus E$, and study the statement? I am having hard time getting started with this problem.
I am after only for hints, not full solution.
Let $\tau =n1_E + m1_{\Omega \setminus E}$. From its definition, you see that $\tau$ can only take two values $$ \tau(\omega)=\begin{cases}n \ \text{ if }\ \omega\in E\\ m \ \text{ otherwise}\end{cases}$$ So $\tau$ is a random variable that takes value $n$ if event $E$ happens, and take value $m$ in any other case.
You want to show that it is a stopping time w.r.t. the filtration $(\mathcal F_k)$, i.e. you want to show that $$\left\{\tau = k\right\}\in \mathcal F_k \ \ \forall k\ge0$$
To prove it, I suggest you treat separately different possible values of $k$, mainly $[k=n]$ (1), $[k=m]$ (2) or $[k\ne n \text{ and } k\ne m]$ (3).
Using the definition of $\tau$, you should be able to compute the events $\left\{\tau = k\right\}$ in all of these cases, and from there you should be able to conclude that $\tau$ is indeed a stopping time.