Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be a continuous and coercive function. Show that if $K$ is a non-empty closed set of $\mathbb{R}^n$, then there exists $x^{\star} \in K$ such that $f(x^{\star}) = \inf_{x \in K} f(x)$.
Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be a continuous and coercive function. Show that if $K$ is a non-empty closed set of $\mathbb{R}^n$, then there exists $x^{\star} \in K$ such that $f(x^{\star}) = \inf_{x \in K} f(x)$.