Every infinite cyclic group has non-trivial proper subgroups

[Mr Davis 97]

Homework Statement

Every infinite cyclic group has non-trivial proper subgroups

Homework Equations

The Attempt at a Solution

I know that if we have a finite cyclic group, it only has non-trivial proper subgroups if the order of the group is not prime. But I'm not sure how to make this argument with infinite groups

 

[fresh_42]

What is a infinite cyclic group and how does it differ from a finite one?

 

[Mr Davis 97]

What is a infinite cyclic group and how does it differ from a finite one?

Doesn't it differ by the fact that there exists no integer $n$ for which $a^n=e$, where $a$ is an element of the group besides e, the neutral element?

 

[fresh_42]

Doesn't it differ by the fact that there exists no integer $n$ for which $a^n=e$, where $a$ is an element of the group besides e, the neutral element?

Yes. So we have $C = \langle a^n\,\vert \,n \in \mathbb{Z} \rangle = \{\ldots , a^{-2},a^{-1},e,a,a^2,\ldots\}$.
Non-trivial subgroup $U$ means $\{e\}\subsetneq U \subsetneq C$.
So what can we conclude from $\{e\}\subsetneq U$?
What could be a subgroup?