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Moderate

Operations on Cauchy Sequences in $\mathbb{R}$

TOPO-XPYV61

Let $(X,\mathcal{T})$ be a topological vector space. In particular, 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 $(x_n)_n$ be a Cauchy sequence in $(X,\mathcal{T})$, and let $(\alpha_n)_n$ be a Cauchy sequence in $\mathbb{R}$, equipped with the usual Euclidean topology.

The answer choices below form mathematical arguments. Find the INCORRECT statements. Select ALL that apply.

A

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 $x_n-x_m\in B$.

B

If $X$ is a normed vector space then $(x_n)_n$ is a bounded sequence.

C

It must be that $(\alpha_n)_n$ is a convergent sequence in $\mathbb{R}$.

D

It must be that $\lim_n \alpha_n=0$.

E

If $X$ is a normed vector space and $\lim_n \alpha_n =:\alpha$, then $((\alpha_n -\alpha) x_n)_n$ must converge to zero.

F

If $\lim_n \alpha_n =:\alpha$, then $(\alpha_n x_n)_n$ must be a convergent sequence in $X$.