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.