well, what you want to do, is show that can be expressed as a positive power of a. and you have almost done done this, which power of a is it?

also, you need to flesh out the existence of m a bit more. all we're given to work with, is that <a> is finite.

i would argue thusly: there are infinitely many positive powers of a, but since G = <a> is finite, for some r ≠ s we must have .

without loss of generality, we may assume that r > s. then . this shows that for SOME positive integer n, namely n= r-s,

we have . we can then let m be the smallest positive integer such that (because we know there is at least one).

this gives us a set of DISTINCT positive powers of a . your proof that ANY positive power of a is in this set consists mainly of the word: "etc."

you can do better. write an arbitrary positive number n, as n = qm + r, where 0 ≤ r < m (why can we do this?). show that .

if is in this set, then so is (why?)