Geometric Sequence with Coprime Extremes is in Lowest Terms/Proof 1

Proof
Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be natural numbers in geometric sequence such that $a_0 \perp a_n$.

there were to exist another set of natural numbers in geometric sequence:
 * $G\,'_n = \sequence {b_0, b_1, \cdots, b_n}$

with the same common ratio where:


 * $\forall k \in \N_{\le n}: a_k > b_k$

From :
 * $a_0 : a_n = b_0 : b_n$

But by hypothesis:
 * $a_0 \perp a_n$

and so from:

and:

it follows that:
 * $a_0 \divides b_0$

However, this contradicts the assumption that $b_0 < a_0$.

Therefore $a_0, a_1, \cdots, a_n$ are the least of those with the same common ratio.