is there something wrong in this question? "Show that if a, b are positive integers, then there is smallest positive integer a−bk, k an integer"

[Dr.Peterson]

let me rephrase what i mean with symbols and tell me where i'm wrong

the well-ordering property is apply only to a subset

if $\displaystyle N$ is the set of natural numbers and $\displaystyle P$ is the set of positive integers with form $\displaystyle a - bk$

then $\displaystyle P$ is a subset of $\displaystyle N$

since $\displaystyle P$ is a subset now, the well-ordering property can be apply

I'd make it a little more precise: The well-ordering property of the natural numbers (positive integers) says that any non-empty subset of the positive integers has a least element; since your set P is such a subset, it has a least element. (That is, it's important to specify what you mean by "apply".)

But have you shown that P is non-empty?

easy
$\displaystyle a-bk>0$
$\displaystyle a>bk$
$\displaystyle \frac{a}{b}>k$

i'm trying to finish the proof with the the well-ordering property

What does this tell you about applying the property?

 

[Steven G]

$\displaystyle a-bk>0$
$\displaystyle a>bk$
$\displaystyle \frac{a}{b}>k$

i'm trying to finish the proof with the the well-ordering property

So k<a/b. Think carefully, real carefully what this is telling you.
You are given that a and b are positive integers. You want to show something about a and b. You think that a and b are arbitrary fixed positive integers and try to prove the theorem.