Repeating for my exam in commutative algebra.
Let G be a topological abelian group, i.e. such that the mappings $+:G\times G \to G$ and $-:G\to G$ are continuous. Then we have the following Lemma:
Let H be the intersection of all neighborhoods of $0$ in $G$. Then $H$ is a subgroup.
The proof in the books is the following one-liner: "follows from continuity of the group operations". (this is from "Introduction to Commutative Algebra" by Atiyah-MacDonald)
I must admit that I don't really see how that "follows". If there is an easy explanation aimed at someone who has not encountered topological groups in any extent, I'd be happy to read it.