Let $m \in \mathbb{N}^{\star}$ and $(u_1, \ldots, u_m)$ be a family of vectors of $\mathbb{R}^n$. We denote $$C = \left\{ \sum_{i=1}^{m} \mu_i u_i, \mu_i \geqslant 0 \; \forall i \in \llbracket 1, m \rrbracket \right\}.$$ Lemma 1 states: If $v \in \mathbb{R}^n$, then one and only one of the following two assertions is verified: (i) $v \in C$, (ii) there exists $w \in \mathbb{R}^n$ such that $\langle v, w \rangle < 0$ and $\langle u_i, w \rangle \geqslant 0$ for all $i \in \llbracket 1, m \rrbracket$. Conclude the proof of Lemma 1.
Let $m \in \mathbb{N}^{\star}$ and $(u_1, \ldots, u_m)$ be a family of vectors of $\mathbb{R}^n$. We denote
$$C = \left\{ \sum_{i=1}^{m} \mu_i u_i, \mu_i \geqslant 0 \; \forall i \in \llbracket 1, m \rrbracket \right\}.$$
Lemma 1 states: If $v \in \mathbb{R}^n$, then one and only one of the following two assertions is verified: (i) $v \in C$, (ii) there exists $w \in \mathbb{R}^n$ such that $\langle v, w \rangle < 0$ and $\langle u_i, w \rangle \geqslant 0$ for all $i \in \llbracket 1, m \rrbracket$.\\
Conclude the proof of Lemma 1.