The following doubt regards a step of a proof of consistency of the maximum likelihood estimator in the case of independent NON identically distributed random variables.
Assume the parameter space is given by $\Omega \subset \mathbb{R}$, denote the maximum likelihood estimator on a sample of $n$ INID random variables $Y_1, \dots, Y_n$, that have probability densities respectively $f_1(y; \theta) , \dots, f_n(y;\theta) $, by $\hat{\theta}_n$ and the true parameter value $\theta_0$ .
The maximum likelihood estimator $\hat{\theta}_n$ in this framework is taken to be the value that maximizes (for simplicity let's assume it exists) $$\ln \prod_{i= 1 }^n f_i(Y_i; \theta)$$
Define $\Omega(\eta) := \Omega \setminus B(\theta_0, \eta) $ where $B(a,r)$ is the open ball centered at $a$ of radius $r$ and $\eta > 0$. Define $$R_n^* = \sup_{\theta} \{ \ln \prod_{i= 1 }^n [ f_i(Y_i; \theta) / f_i( Y_i; \theta_0) ] : \theta \in \Omega(\eta) \}$$
I want to prove that $\{ \hat{\theta}_n \in \Omega(\eta) \} \subset \{ R_n^* \ge 0 \}$ , how can this be seen?
To write my comment as an answer: Define $g:\Omega\rightarrow\mathbb{R}$ by $$ g(\theta) = \log \left(\prod_{i=1}^n \frac{f_i(Y_i;\theta)}{f_i(Y_i;\theta_0)}\right)$$ By definition of $\hat{\theta}_n$ as the global maximizer of $g(\theta)$ over all $\theta \in \Omega$ (and since $\theta_0\in \Omega$) we have $$ g(\hat{\theta}_n) \geq g(\theta_0)=0$$ Now $R_n^*$ is the maximum of $g(\theta)$ over a restricted domain. But if $\hat{\theta}_n$ is already in that restricted domain, then $R_n^*=g(\hat{\theta}_n)$ and hence $R_n^*\geq 0$.