Every number n > 6 is sum of two numbers $a$ and $b$ with $a,b>1$ and $gcd(a,b)=1$

Question

I was working on this question and came up with this answer: Take any prime number $p$ that’s greater than $\frac{n}{2}$.

Let $a=p$ and $b=n-p$, then b is necessarily smaller than a because $p>\frac{n}{2}$ and $gcd(a,b)=1$, because $p$ is prime.

Then we found $a$ and $b$ with $a+b=n$.

Does this count as a prove or is there a smarter way? Got an exam on Monday and appreciate any help!