I've recently starting reading Discrete Mathematical Structure - KBR. One of the exercises asks the following:
Prove that if $a,b \in \mathbb{Z}^+$, that $a|b$ and $b|a$, then $a=b$.
My solution (direct proof):
$a|b \rightarrow b=k_1 a \\ b|a \rightarrow a=k_2 b$
It follows from the above then that:
$ a = k_2(k_1a) \\ \Updownarrow \\ 1 = k_1k_2 $
Therefore
$b=k_1 a \Leftrightarrow b=1a \Leftrightarrow b=a$
QED.
Is this considered correct, and is the notation allright? Thanks!
Soo I guess there's multiple ways to go about proving the above question. The simplest would however be to consider the comment left by Peter in the comments:
$a|b \rightarrow a \leq b \\ b|a \rightarrow b \leq a$
Considering the fact that $a,b$ are both positive it follows that $a=b$.