Generator of a group cannot be a generator of a proper subgroup

Question

While solving problem on cyclic group, I have noticed that Generator of every proper subgroup of $\mathbb{Z_n}$ cannot generate the whole $\mathbb{Z_n}$ or equivalently, generator of $\mathbb{Z_n}$ cannot generate a proper subgroup of $\mathbb{Z_n}$.
I try to prove this assumption.
Let m be a order d generator of a proper subgroup of $\mathbb{Z_n}$. So $n=dk$ for some integer $k$ and $d<n$. Therefore $<m>=\{0, m,\ldots, (d-1)m\}$. Now $(d-1)<(n-1)$, so $|<m>|\neq n=|\mathbb{Z_n}|$.
Now my question is whether this assumption and the supporting reasoning are correct. Also, if the assumption is correct, can we say in general that Generators of a cyclic group cannot generate a proper subgroup? Thank you.