Jensen's Inequality for Finite Convex Combinations in $\mathbb{R}^n$

In the setting of Convex Functions on $\mathbb{R}^n$: Standing Notation, let $C\subseteq\mathbb{R}^{n}$ be convex and let $f:C\to\mathbb{R}$ be convex. Let $N\in\mathbb{N}$, let $x_{1},\dots,x_{N}\in C$, and let $t\in\Delta_{N}$ be convex weights, so that the convex combination $\sum_{k=1}^{N}t_{k}x_{k}$ lies in $C$ by Convex Combinations: Small Cases, Splitting Off the Last Point, Dropping a Zero Weight, Joining Two Combinations, and Membership in Convex Sets (vi).

(i) Jensen's inequality.

$$f\Bigl(\sum_{k=1}^{N}t_{k}x_{k}\Bigr)\le\sum_{k=1}^{N}t_{k}f(x_{k}).$$