Non cyclic subgroup of $\mathbb{Z}_5$

Question

Let $H = \{0, 2, 3\} \subset \mathbb{Z}_5$. $H$ is a subgroup of $\mathbb{Z}_5$, since it is closed with respect to addition and with respect to inverses. Given that $\langle 2 \rangle = \langle3\rangle = \mathbb{Z}_5 \ne H$, $H$ is not generated by any of its elements. Therefore, $H$ is not cyclic, contradicting the theorem that "every subgroup of a cyclic group is cyclic".

Now, there is a foolish and obvious mistake in the reasoning above, but I can't figure out what it is. Can anybody help me?

Answer 1

$2+2=4$ ${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$

Answer 2

H is not closed under addition