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Topology

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Operations on Cauchy Sequences in a Topological Vector Space

TOPO-WXSVLY

Let $(X,\mathcal{T})$ be a topological vector space. This means that the vector addition and the scalar multiplication are continuous functions from $X\times X$ to $X$ and from $\mathbb{R} \times X$ to $X$, respectively. Let $(x_n)_n$ and $(y_n)_n$ be Cauchy sequences in $(X,\mathcal{T})$.

Which of the following MUST be true? Select ALL that apply.

A

$(2x_n)_n$ is a Cauchy sequence in $X$.

B

$(2 x_n)_n$ is a convergent sequence in $X$.

C

$(x_n+y_n)_n$ is a Cauchy sequence in $X$.

D

$(x_n)_n$ is a bounded sequence in $X$.

E

$(2x_n-3y_n)_n$ is a Cauchy sequence in $X$.

F

For any open neighborhood $B$ of the zero vector, there exists some $n_0\equiv n_0(B)\in \mathbb{N}$, such that for all $n,m\geq n_0$ we have $x_n-x_m\in B$.