Is every positive integer (eventually) a sum of $m$ relatively prime integers for every $m$?

Question

Is this a theorem?

Given any integer $m$, there exists $N(m)$, such that for all $n>N(m)$, we have $$n = a_1+a_2+\cdots+a_m$$ where $\gcd(a_i,a_j) = 1$ with $a_i>1$ for all $i,j>1$.

For instance, it is a textbook exercise to show for small values of $m$; Any integer exceeding $6$ is a sum of two relatively prime numbers; Any integer exceeding $17$ is a sum of three relatively prime numbers and so on.

Answer 1

We may induct on $m$. Assume that for $m-1$ this is proved. Choose $a_1$ as a prime number between $n/2$ and $n-N(m-1)$ and apply induction proposition for $n-a_1$.