The claim is the following: Let $V$ be a finite-dimensional real vector space and let $C \subset V $ be convex and non-empty such that $0 \notin C$. Then there exists some $l \in \operatorname{Hom}(V,\mathbb{R})$ such that $l_{\vert C} \ge 0$ and $l(C) \neq \{0\}$.
Proof. Induction over $\dim V$: For $\dim V = 1$ the claim is clear. Let $\dim V = n+1$. Then let $H$ be an n-dimensional hyperplane intersecting $C$. Set $D = H \cap C$. By induction hypothesis, one has a $u \in\operatorname{Hom}(H,\mathbb{R})$ with $u_{\vert D} \ge 0$ and $u(D) \neq \{0\}$. Now let $p_H : V \rightarrow H$ be the canonical projection. Now define $l := u \circ p_H$.Then one has $l_C \ge 0$ and $l(C) \neq \{0\}$.
For some reason, I think this proof is incorrect, but I can't spot the mistake.
The proof is correct. It is so simple because it is only giving to you that $u$ is nonzero on some elements of $C$, but it will be zero on many. If you take $H$ to "barely" cross $u$, then $u$ will be zero on most of $C$.
The proof would be harder if you were trying to find a $u$ that is nonzero for all but one point of $C$.