Currently I am trying to understand the proof of the following lemma which appear in the book Additive combinatorics of Tao and Vu (Chapter 5, page 212, in the 2006 edition).
Lemma 5.14 Let $A, B$ be additive sets in a finite-dimensional vector space $V$ such that $A$ and $B$ both contain $0$, and suppose that $0$ is a vertex on the convex hull of $A \cup B$. Let $A^{\prime} := A \setminus \{0\}$ and $B^{\prime}:= B \setminus \{0\}$, and $C := (A^{\prime} \cup B^{\prime} )\setminus(A^{\prime} + B^{\prime})$. Then $A + B=\lbrace a+b:a\in A, b\in B\rbrace$ lies in the subspace of $V$ spanned by $C$.
In the proof of the theorem the book start the proof as follows
Proof
Without loss of generality we may take $V = \mathbb{R}^n$ . By the Hahn–Banach theorem, there exists a linear functional $\varphi : V → \mathbb{R}$ such that $\varphi(x) > 0$ for all $x \in (A \cup B)\setminus \{0\}$. We need to show that every element $x$ of $A + B$ lies in the span of C. We shall prove this by induction on $\varphi(x)$, which is a non-negative integer.
In this first part of the proof I have two questions.
Question 1. Why we can assume that $\varphi(x)$ is an integer ? My ideas are the following. Since $\varphi:\mathbb{R^n}\to \mathbb{R}$ we already know that $\varphi$ should have the form $\varphi(x_1,\ldots, x_n)=\sum_{j=1}^n \alpha_jx_j$ for $\alpha_j\in \mathbb{R}$ and from here I think that we can do a suitable transformation in the $\alpha_j$ to obtain an integer (but this does not clear to me).
Question 2. How to apply the Hanh-Banach theorem in this case? This is the principal problem. I understand that the theorem say that we can extend a linear functional $g:M\to \mathbb{R}$ dominated by a seminorm $\psi:V\to \mathbb{R}$ to a linear functional $f:V\to \mathbb{R}$ such that $f\mid_M=g$ and $f\leq \psi$ for all $x \in V$. Frome here I do not understand which subspace $M$ is taken.
Any hints or answers are welcome.