Finite Sums of Cᵏ Functions Are of Class Cᵏ

In the setting of Functions of Class Cᵏ: Notation, let $k\in\mathbb{N}_{0}$ and $N\in\mathbb{N}$, and let $f_{1},\dots,f_{N}:U\to\mathbb{R}$.

(i) Functions of class $C^{k}$. If $f_{i}$ is of class $C^{k}$ on $U$ for every $i\in[N]$, then the function $U\to\mathbb{R}$, $x\mapsto\sum_{i=1}^{N}f_{i}(x)$, whose value at $x$ is the finite sum of the real numbers $f_{1}(x),\dots,f_{N}(x)$, is of class $C^{k}$ on $U$.

(ii) Smooth functions. If $f_{i}$ is smooth on $U$ for every $i\in[N]$, then the function $U\to\mathbb{R}$, $x\mapsto\sum_{i=1}^{N}f_{i}(x)$, is smooth on $U$.