Extreme Value Theorem

Thm. (EVT)
If $f: K \to \mathbb{R}$ is conti. on a compact $K \subseteq \mathbb{R}$, then $f$ attains a maximum and minumum value.

Proof.
Since $K$ is compact, $f(K)$ is compact as well.
We can set $\alpha = \sup f(K)$ since $f(K)$ is bounded and also one can observe that $\alpha \in f(K)$ due to the closedness of $f(K)$.
Thus $\exist x_1 \in K$ such that $\alpha = f(x_1)$.

Thm.
$\Theta_0 = (a, b) \subseteq \mathbb{R}$ : open set
$\ell \in C^2(\Theta_0)$ with 1) $\nabla^2 \ell(\theta) <0$ for any $\theta \in \Theta_0$ and 2) $\lim_{\theta \to \partial \Theta_0} \ell(\theta) = -\infty$.
Then, $\exist ! \hat \theta \in \Theta_0$ such that $\ell(\hat \theta) = 0$.

Sketch of the proof.
1. EVT를 통해서 $\hat \theta$가 $\Theta_0$에서 $\ell$의 최대임을 보이자. (Compact Set을 잘 잡는게 킥...?)
2. Strictly Concave하다는 것이 조건으로 부터 자동으로 도출되기 때문에, 이를 이용해서 유일한 해가 존재함을 보일 수 있다.