Proof by well ordering: Every positive integer greater than one can be factored as a product of primes. Part II

Question

I have a question related to the one found here. I am struggling to understand this part of the proof:

So $n$ must be a product of two integers $a$ and $b$ where $1 < a, b < n$. Since $a$ and $b$ are smaller than the smallest element in $C$

How come $1 < a$ and $b < n$? Where does it come from? And why does it imply that $a$ and $b$ are smaller than the smallest element in $C$ if $C$ is by definition contains all the nonnegative integers greater than $1$?

Upd.: this is not a duplicate of Proof by well ordering: Every positive integer greater than one can be factored as a product of primes. because in this question there is a different statement that needs to be explained, hence Part II in the heading.

...and you can't have an integer factor of $n$ be bigger than $n$. — David G. Stork, Jul 05 '18 at 20:44
@m_t_ still don't get it... Please, try explaining as if I were 10 years old. — urbanchef, Jul 05 '18 at 20:59
@DavidG.Stork Ok, then why a < n condition is not mentioned in the proof since it is one of the factors of n? But it is said that 1 < a? — urbanchef, Jul 05 '18 at 20:59
I suspect the original notation (;) was to really mean $1 < a \leq b < n$. — David G. Stork, Jul 05 '18 at 21:03
@DavidG.Stork Thanks! It all adds up now based on the fact that an integer factor of n can't be bigger than n itself. And n is strictly greater than both a and b because otherwise n would be a prime which is not possible by the definition of C set. — urbanchef, Jul 05 '18 at 21:19
@AaronMeyerowitz not only those, but also the ones that cannot be factored as a product of primes. Thank you! — urbanchef, Jul 05 '18 at 21:33
An integer $n > 1$ is $n = 1*n$ so all integers $> 1$ have at least two naturel factors. The primes have exactly two and the non-primes have more than two. So if $n>1$ is not prime it has a factor $a; a\ne 1;a\ne n$. $a < 1$ is impossible as $a$ is natural. $a>n$ is impossible as being a factor means there is natural number $b$ so that $ab=n$. As $k \ge 1$ if $a>n$ then $ab>n$ which is a contradiction. So $1<a<n$. Now the $b$ so that $ab=n$ is also a factor and by the exact same argument $1<b<n$. — fleablood, Jul 05 '18 at 23:26
Possible duplicate of Proof by well ordering: Every positive integer greater than one can be factored as a product of primes. — amWhy, Jul 06 '18 at 00:12

score 1 · Answer 1 · answered Jul 05 '18 at 21:59

Sorry I deleted my comment. The proof attempts to show $C$ has no members at all. It is defined as only the integers greater than $1$ which are not prime and do not factor into primes. Let $D$ be the numbers starting at $2$ which do factor into primes or are prime themself. If two numbers are in $D$ so is their product. If, in fact, $C$ is not empty it has a smallest member $n$ so if $1 \lt m\lt n$ then $m$ is in $D$ but not in $C$ and does factor into primes. But for this smallest $n$ in $C$ , $n$ is not prime so $n$ does factor into $n=ab$ with $2 \leq a \leq b \leq \frac{n}2$ so neither of $a$ or $b$ is in $C$, they are both in $D$ as is their product $n=ab.$ that is a contradiction.

Proof by well ordering: Every positive integer greater than one can be factored as a product of primes. Part II

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