Convex Subset of $\mathbb{R}^n$

Let $n\ge1$ be a natural number, let $\mathbb{R}$ be the ordered field of real numbers, and regard Euclidean space $\mathbb{R}^n$ as a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, with the sum $z+z'$ of points and the scalar multiple $\lambda z$.

A subset $C\subseteq\mathbb{R}^n$ is convex if for all $x,y\in C$ and every $t\in\mathbb{R}$ with $0\le t$ and $t\le 1$,

$$t\,x+(1-t)\,y\in C .$$