Let $G=\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times ...$. We define $+:G\times G\rightarrow G$, where $+$ is defined as sumation on each component via module $2$. Let $n\in \mathbb{N}$ and $a_1,...,a_n\in G$. Prove: $G\neq \langle a_1,...,a_n\rangle$.
I was able to prove that $(G,+)$ is a group. Even more, I notice that $(G,+)$ is an Abelian group where order of each element is $\leq 2$. Now, since I want to prove $G\neq \langle a_1,...,a_n\rangle$, it is enough to show that there exists $g \in G$, such that $g\notin\langle a_1,...,a_n\rangle$. Here I didn't know what $g$ should I pick. Any idea?
