Assume $X_1, ... , X_n$ statisfy the regularity conditions, where $\theta_0$ is the true parameter, and further that $f(x,\theta)$ is differentiable with respect to $\theta$ in $\Omega$. then the likelihood equation, $\frac{\partial}{\partial\theta}L(\theta)=0$, or equivalently $\frac{\partial}{\partial\theta}l(\theta)=0$, has a solution $\hat{\theta}_n$ such that $\hat{\theta}_n \rightarrow \theta_0$ in probability.
This is how the proof starts:
Because $\theta_0$ is an interior point in $\Omega$, $(\theta_0-a,\theta_0+a) \subset \Omega$, for some $a>0$. Define $S_n$ to be the event
$$S_n = \{X: l(\theta_0;X)>l(\theta_0-a;X)\} \cap \{X: l(\theta_0;X) > l(\theta_0+a; X)\}$$
I'm having a hard time understanding how they came up with $S_n$. For example, if we have an interval $(a,b)$...then if we look at the set $\{x:x>a\} \cap \{x:x>b\}$, we would get $\{x:x>a\} \cap \{x:x>b\}$ = $\{x:x>b\}$. So I don't really understand how they got $S_n$.
Thanks in advance
