Can the following lemma be proved?

Lemma (Rokach-Goldstein)

Let $x_i$ be a generalized Fibonacci sequence of positive integer numbers, $x_0,x_1,x_2,\ldots$ such that for every $i\ge2$, we have $x_i=x_{i-1}+x_{i-2}$, and where $x_0$ and $x_1$ are coprime to each other.

In this case, there exists some $x_n$ , such that $x_n$ is coprime to the sum of all numbers in the sequence.