Jensen's Inequality for Finite Convex Combinations

Let $n$ and $N$ be natural numbers with $1\le n$ and $1\le N$, let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure, and let $[N]$ be the initial segment determined by $N$. Let $C$ be a convex subset of Euclidean space $\mathbb{R}^n$, regarded as a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, and let $f:C\to\mathbb{R}$ be convex on $C$.

Let $x:[N]\to\mathbb{R}^n$ take all of its values in $C$ and let $t$ be a system of convex weights of length $N$. The convex combination $\sum_{k=1}^{N}t_kx_k$ lies in $C$ by claim 3 of Small Cases, Reduction, and Membership for Convex Combinations, and

the sum on the right being a finite sum in the field $\mathbb{R}$.