Part of a lemma from a book "Diophantine Inequalities" by R.C. Baker I am going through lemma 3.1 from "Diophantine Inequalities"
by R.C.Baker and I was wondering if someone could
possibly give me a hint to figure out this one part...
We have the following:
$\rho > 0$, and $\gamma_1, \gamma_2, ... \ \gamma_R$ be reals with
$$
|| \gamma_i - \gamma_j|| \geq \rho
$$
for $i \not = j$.
WLOG assume 
$$
||\gamma_1|| \leq ... \ \leq || \gamma_R||.
$$
I am trying to get that $||\gamma_i|| \geq(i-1)\rho/2$
for $i = 2, 3, ... \, R$. And by $||x||$,
I mean $$ ||x|| = \min_{n \in \mathbb{Z}} |x-n|.$$
I could do the case $i=2$.
Could someone please give me a hint on how to
get the cases when $i > 2$?
Thank you very much.

Let's prove the contrapositive: given $\rho > 0$ and $||\gamma_1|| \leq \cdots \leq || \gamma_R||$, if there exists an $i$ such that $||\gamma_i|| < (i-1)\rho/2$, then there exist $i\ne j$ such that $|| \gamma_i - \gamma_j|| < \rho$.
Without loss of generality, assume that all $\gamma_k$ are between $-\!\frac12$ and $\frac12$. By assumption, $-(i-1)\rho/2 < \gamma_1, \dots, \gamma_i < (i-1)\rho/2$. Either at least half of the $\gamma_i$ are nonnegative or at least half are nonpositive. Say at least half are nonnegative; then (reordering them conveniently) we have $0 \le \gamma_1, \dots, \gamma_{\lceil i/2 \rceil} < (i-1)\rho/2$; then the pigeonhole principle should imply that some $\gamma_i-\gamma_j$ is small enough.
