GCD of Generators of General Fibonacci Sequence is Divisor of All Terms

Theorem
Let $\mathcal F = \sequence {a_n}$ be a general Fibonacci sequence generated by the parameters $r, s, t, u$:


 * $a_n = \begin{cases}

r & : n = 0 \\ s & : n = 1 \\ t a_{n - 2} + u a_{n - 1} & : n > 1 \end{cases}$

Let:
 * $d = \gcd \set {r, s}$

where $\gcd$ denotes greatest common divisor.

Then:
 * $\forall n \in \Z_{>0}: d \divides a_n$

Proof
From the construction of a general Fibonacci sequence, $a_n$ is an integer combination of $r$ and $s$.

From Set of Integer Combinations equals Set of Multiples of GCD, $a_n$ is divisible by $\gcd \set {r, s}$.

Hence the result.