There is something wrong with your proof. You claim that since there are only $n$ primes, there are only $n!$ composite numbers, which is not true. Even a single prime, $2$ for example, produces infinitely many composites:
$$2,4,8,16,32,\dots, 2^n, \dots$$
The problem is that you did not state that those are equivalences (usually denoted by $\iff$) between your lines. And you do not even need the equivalences, you only need the implicatiosn from the bottom to the top, so you should perhaps write your proof "upside down".
The way your proof is presented right now makes it look like the top implies the bottom. (Which it does) but that doesn't mean that the bottom implies the top. So I'd write down:
For any positiv integer $n$ the following inequality obviously holds:
$$2n > n$$
This implies
$$n^2-n^2+2n > n^2-n^2+n$$
etc.
Notice that this is also the way you'd want to read your proof so that people understand it. And it is important to realize that this is not necessarly the way you found the proof.
Whenever you find a proof somewhere you can be quite sure that the way it is presented to you has nothing to do with how the one who proved it found the proof. It is really just written down nicely in order for the reader to be able to nicely follow the chain of arguments.
But don't worry, writing "nice" proofs does take a while at the beginning of your math career=)
Regarding the comment: You could prove the inequality by contradicion, e.g. suppose that the inequality does not hold, and then find a contradiction, but this is not necessary here.
EDIT: It seems to me that you are unfamiliar with the concept of logical implications. If two mathematical statements $A,B$ (e.g. equations, inequalities etc.) mean the same thing, they are called equivalent, denoted by the bidirectional double arrow. $$A \iff B$$
Example: Let $x$ be a real number. Then following equivalence holds: $$x -5 = 0 \iff x = 5$$
But if $A$ only hold if $B$ holds, then we say $A$ implies $B$ or alternatively, "if $A$ holds then $B$ must hold too." This is denoted by a simple doublearrow: $$A \implies B$$
Example: Let $x$ be a real number. Then following implication holds:
$$x = 5 \implies x^2 = 25$$ But "the other way around" is not necessarily true, as the right statement is also true for $x=-5$ but not the left one.
Best Answer
I'm assuming you're in a number theory class or abstract algebra. At this level of such a course, we haven't formally reintroduced $\Bbb Q $ so fractions don't formally exist yet.
We have multiplication and addition. And we have the integers.
The better proof is to show that $\gcd(18,6)=6$ and hence that the smallest positive linear combination of $18$ and $6$ we can make is $6$.