Elements of Geometric Sequence from One which Divide Later Elements

Theorem
Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers.

Let $a_0 = 1$.

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

Then:
 * $\forall r \in \set {0, 1, \ldots, m}: a_k \divides a_m$

where $\divides$ denotes divisibility.

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

for some $q \in \Z$.

Hence the result from Divisors of Power of Prime.