Moderate# Cauchy Sequences and Functionals

TOPO-FNWOYP

Let $(X,\mathcal{T})$ be a topological vector space. Assume that $\mathbb{R}$ is equipped with the usual Euclidean topology. Let $f:X\to \mathbb{R}$ be a map such that for any $a,b \in X$, we have $|f(a) - f(b)| \leq | f(a-b)|. $ Let $e$ be the zero vector in $X$. Assume that $f(e)=0$, and that $f$ is continuous at $e$. Let $(x_n)_n$ be a Cauchy sequence in $(X,\mathcal{T})$.

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