Do infinite coprime points $(x,y)$ exist satisfying $x+y=n$ where $x,y,n \in \mathbb{N}$?

Question

I'm trying to prove infinite primes exist based upon this:
For $n$ prime, every grid point on the line is coprime.

Examples of $x+y=n$ for $n$ prime

n=3  {(1,2),(2,1)}
n=5  {(1,4),(2,3),(3,2),(4,1)}
n=7  {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}
...

Can it be shown there exists infinite such sequences, thus proving infinite primes exist?

Thanks.