I'm currently starting a number theory book. On its exercise, there's:
Prove: If $a,b \in \mathbb{N}$ and $ab=1$, then $a=1$ and $b=1$.
Here's one proof I just did:
Since $ab=1$, $a=\frac{1}{b}$. But note that if $b \neq 1$, $b > 1$, and thus $\frac{1}{b} \notin \mathbb{N}$, which contradicts $a \in \mathbb{N}$. Similarly for $b$.
Is it safe to assume if $b \neq 1$, $b > 1$, and thus $\frac{1}{b} \notin \mathbb{N}$?
Also, here's another proof:
Since $ab=1$, $\forall c \in \mathbb{N}$, $abc=1c=c$.
Note that since $a \in \mathbb{N}$ and $c \in \mathbb{N}$, $ac \in \mathbb{N}$.
Thus, $\frac{abc}{b} = \frac{c}{b} = ac \in \mathbb{N}$.
Since $\frac{c}{b}$ for all $c \in \mathbb{N}$, $b=1$.
Are those 2 proofs OK? I'm wondering how far can I assume things to be "axioms"?