Is $\mathbb{Z}\oplus\mathbb{Z}$ cyclic? I know there are infinitely many proper subgroups of $\mathbb{Z} \oplus \mathbb{Z}$, but is $\mathbb{Z} \oplus \mathbb{Z}$ cyclic?
Take a and b different integers.

Easy way to think is 

If there is any generator, it can be either of the form
1)(a,a) or 2)(a,b)

If it is of form 1, you can never obtain element of second form.

Can you finish the argument?
Let's use @Arturo Magidin's hint:  $\Bbb Z\oplus\Bbb Z$ has the nontrivial subgroup $\Bbb Z\oplus\{e\}$, which is not of finite index.

$\therefore \Bbb Z\oplus\Bbb Z$ is not cyclic.

Reasoning behind the hint:   an infinite cyclic group is naturally isomorphic to $\Bbb Z$, whose only subgroups are of the form $n\Bbb Z$.