Joined: Tue Nov 10, 2015 8:25 pm Posts: 313

Definition: Let $G$ be a topological abelian group (written additively) such that $0 \in G$ has a countable fundamental system of neighborhoods. A Cauchy sequence in $G$ is defined as a sequence $(x_{n})_{n \in N}$ of elements of $G$ such that for every neighborhood $U$ of $0$ there exists an integer $s = s(U) $ such that $ x_{n}  x_{m} \in U $ for all $n,m \geq s$.
Let $ f \ \colon G \longrightarrow H $ be a continuous homomorphism between topological abelian groups (written additively). Show that $f$ preserves Cauchy sequences, i.e. the image of a Cauchy sequence in $G$ under $f$ is a Cauchy sequence in $H$.

