I dont understand why every sub group of cyclic groups are also cyclic.

Question

This doesn't make sense to me. A group is cyclic if it can be generated by a single element. Let's say that the element is $\langle a \rangle$. since it is cyclic, at one point $a^n$ must be equal to the identity, where $n$ is a positive integer. but if a subgroup is cyclic, then its generator must also be powers of $a$, and $a^n$ must also be equal to the identity. so I don't understand how we can have a smaller sub-group than $G$, and it contains the same identity and still be cyclic if $a^n$ must be equal to the same identity for both groups. this just means that $n$ should be equal in both groups. and if $n$ is equal then you should have the exact same group