🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

Suppose that $C\subseteq {\mathbb{R}}^n$ is compact, and suppose $f:C\to {\mathbb{R}}^m$ is continuous, then the image $f\left(C\right)$ is compact.

Suppose that $C\subseteq {\mathbb{R}}^n$ is compact and $f:C\to \mathbb{R}$ is continuous, then there are points $a,b\in C$ that attain the minimum and maximum values of $f$ on $C$, that is, for every $x\in C$ we have: $${\displaystyle f\left(a\right)\le f\left(x\right)\le f\left(b\right)}$$