Let $E$ be a topological vector space, $A \subseteq E$ and $S:=\{x_1,x_2,x_3,\cdots\}$ a sequence in $A$. I know that, if for each $n\in \mathbb{N}$, we define $$S_n:=\{x_{n+1},x_{n+2},\cdots\}$$ then the family $B:=\{S_n \subset A \; ; \; n \in \mathbb{N}\}$ is a base for the filter $\mathcal{F}_S$ given by $$\mathcal{F}_S=\{B\subset E \; ; \; S_n \subset B, \:\text{for some} \: n \in \mathbb{N}\}.$$
A priori, the filter $\mathcal{F}_S$ is a filter in $E$. It's this filter a filter in $A$ too?
Moreover, if $S$ is a Cauchy sequence in $A$, then $\mathcal{F}_S$ a Cauchy filter in $A$?