How to show that $\Omega_{c}=\left\{x \in \mathbb R^{n}: V(x) \leq c\right\}$ is compact?
Solution 1:
Let $c > 0$. We have to show that $\Omega_c$ is closed and bounded.
$\newcommand{\R}{\mathbb{R}}$
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Since $V$ is differentiable, $V:\R^n \to \R$ is continuous. Therefore, $\Omega_c$ is closed.
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From the definition of radial unboundedness $$ \forall c > 0: \exists r > 0: \forall x \in \R^n : (||x|| > r \Rightarrow V(x) > c) $$ where the contrapositive of the conditional is $$ V(x) \le c \Rightarrow ||x|| \le r $$ Therefore, $\Omega_c$ is bounded.