Easy# 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.