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.

Can the following lemma be proved?

Lemma (Rokach-Goldstein)

Let $x_i$ be a finite generalized Fibonacci sequence of positive integer numbers, $x_0,x_1,x_2,\ldots,x_m$ such that for every $2\le i\le m$, 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 $n$, $0\le n\le m$, such that $x_n$ is coprime to the sum of all numbers in the sequence.

Can the following Lemma be proved?

Lemma (Rokach-Goldstein)

Let $x_i$ be a Generlized 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 co-prime 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.

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.

Can the following Lemma be proved?

Lemma (Rokach-Goldstein)

Let x be a Generlized Fibonacci Sequence of positive integer numbers, x0 , x1 , x2 … such that for every I >=2 , xi = xi-1 +xi-2 , and x0 , x1 are co-prime to each other.

In this case  , there exists some x , such that x is co prime to the sum of all numbers in the sequence.

Can the following Lemma be proved?

Lemma (Rokach-Goldstein)

Let $x_i$ be a Generlized 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 co-prime 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.