union of a chain of groups as group

1. Proposition

Let $G₁ \subseteq G₂ \subseteq G₃ … $ be a chain of groups. Then $G \coloneqq \bigcup_{n \in \mathbb{N}} G_n$ is also a group

Let $g_1 \in G$, then by assumption there exists a $G_i$ such that $g_1 \in G_i$ and hence $g_1^{-1} \in G_i \subseteq G$