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 Post subject: Continuous Homomorphism And Cauchy SequencesPosted: Tue Jun 21, 2016 6:37 pm
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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 $H$.

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