Let $V$ be a vector space, and $W$ an irreducible, crystallographic Coxeter group on $V$ with simple root system $\Pi = \{a_1 \cdots a_n \}$. We define a partial ordering on $V$ by $u \geq v$ iff $u - v = \sum_{i = 1}^{n} k_i a_i$ with $k_i \geq 0 \forall i$. Let $x \in V$ such that the dot product of $x$ with any simple root is positive: $(x/a_i) \geq 0 \forall i$.
I want to prove that $\forall w \in W x \geq w(x)$ w.r.t the partial order defined above. My idea is to do this by induction on the length of $w$: indeed if $w$ is a simple reflection $s_i$ then we have: $$ s_i (x) = x - 2 \dfrac{(x/a_i)}{(a_i/a_i)} a_i $$ and thus $$ x - s_i (x) = 2 \dfrac{(x/a_i)}{(a_i/a_i)} a_i \geq 0 $$
However I am not sure how to proceed to prove the heredity: if I write $w = \tilde{w} s_{i}$ I obtain $$w(x) = \tilde{w}(x) - 2 \dfrac{(x/a_i)}{(a_i/a_i)} \tilde{w}(a_i)$$ But this seems like a dead end since $\tilde{w}(a_i)$ is not necessarily a simple root... Any hint would be very appreciated!
$\tilde{w}(a_i)$ may not be a simple root, but we may assume it is a positive root, otherwise $\tilde{w} s_i$ could be replaced by a shorter expression. This uses the standard result that if $w a_i < 0$ then $l(w s_i) < l(w)$.
Hence $x \ge \tilde{w}(x) \ge w(x)$, the first $\ge$ by induction and the second $\ge$ by the fact that $\tilde{w}(a_i) > 0$ and $(x,a_i) \ge 0$.