im trying to prove that the subgroup of a cyclic group is always cyclic but im wondering what is wrong with my proof.

what i did:

let G=<g>={g^n l n is an integer} and let H be the subgroup with order d. then d is the least positive integer such that g^d=e.

thus d is a divisor of n.

since the order of g^n =G is the order of H =d then d=n and hence H=<g>

im wondering what is wrong with my understanding because in my notes, it states that to proof this theorem, i need to let H=<g^d> ={(g^d)^q} and im wondering why i need to do it that way.

buy letting H =<g^d>, it still means that it has an order d right?