Affine Functions, Sums, Nonnegative Multiples and Pointwise Suprema of Convex Functions

Let $n\ge1$ be a natural number, let $\mathbb{R}$ be the ordered field of real numbers, and let $C\subseteq\mathbb{R}^n$ be convex, where Euclidean space $\mathbb{R}^n$ is a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space.

Then the following hold, convexity of a real-valued function on $C$ being understood in the sense of Convex Real-Valued Function on a Convex Subset of $\mathbb{R}^n$.

1. (Affine functions) Let $p\in\mathbb{R}^n$ and $c\in\mathbb{R}$, and let $\ell:C\to\mathbb{R}$ be given by $\ell(x)=p\cdot x+c$, with the dot product. Then $\ell$ is convex on $C$.

2. (Sums) If $f,g:C\to\mathbb{R}$ are convex on $C$, then so is the function $f+g:C\to\mathbb{R}$ whose value at $x$ is $f(x)+g(x)$.

3. (Nonnegative multiples) If $f:C\to\mathbb{R}$ is convex on $C$ and $\mu\in\mathbb{R}$ satisfies $0\le\mu$, then the function $\mu f:C\to\mathbb{R}$ whose value at $x$ is $\mu\,f(x)$ is convex on $C$.

4. (Pointwise suprema) Let $\mathcal{F}$ be a nonempty set of functions from $C$ to $\mathbb{R}$, each convex on $C$, and suppose that for every $x\in C$ the set $\{f(x):f\in\mathcal{F}\}$ has an upper bound in $\mathbb{R}$. Then the function $F:C\to\mathbb{R}$ whose value at $x$ is the least upper bound of $\{f(x):f\in\mathcal{F}\}$ is well defined, and $F$ is convex on $C$.