Proof of Lemma: Every integer can be written as a product of primes I'm new to number theory. This might be kind of a silly question, so I'm sorry if it is.
I encountered the classic lemma about every nonzero integer being the product of primes in Ireland and Rosen's textbook A Classical Introduction to Modern Number Theory. In this textbook there is also a proof for it provided, and I'd like to understand why it is that the proof actually works. 
The proof is as follows: 

Assume, for contradiction, that there is an integer $N$ that cannot be written as a product of primes. Let $N$ be the smallest positive integer with this property. Since $N$ cannot itself be prime we must have $N = mn$, where $1 < m, n < N$. However, since $m$, $n$ are positive and smaller than $N$ they must each be a product of primes. But then so is $N = mn$. This is a contradiction. 

I feel like this proof kind of presupposes the lemma. I think this line of reasoning could be strengthened using induction, and I've seen other proofs of this lemma that use induction. Can someone help me out? What am I missing and why do I think that this proof of the lemma is circular?  
Edit: I'd like to add that this textbook states that if $p$ is a prime number, then so is $-p$. That's where my confusion stems from. 

Although the proof by contradiction is correct, your feeling of unease is fine, because the direct proof by induction is so much clearer:

Take an integer $N$. If $N$ is prime, there is nothing to prove. Otherwise, we must have $N = mn$, where $1 < m, n < N$. By induction, since $m, n$ are smaller than $N$, they must each be a product of primes. Then so is $N = mn$. Done.

