Elements of Geometric Sequence from One which Divide Later Elements

Theorem
Let $G_n = \left\langle{a_n}\right\rangle_{0 \mathop \le i \mathop \le n}$ be a geometric progression of integers.

Let $a_0 = 1$.

Let $m \in \Z_{> 0}$.

Then:
 * $\forall r \in \left\{{0, 1, \ldots, m}\right\}: a_k \mathop \backslash a_m$

where $\backslash$ denotes divisibility.

Proof
By Form of Geometric Progression of Integers from One, the general term of $G_n$ can be expressed as:
 * $a_j = q^j$

for some $q \in \Z$.

Let $r \in \Z$ such that $0 \le r \le m$.

Then:
 * $a_r = q^r$

But:
 * $a^m = q^r q^{r - m}$

Hence the result by definition of divisibility.