Let $\mathrm{V}$ be a metrizable topological vector space (either over $\Bbb R$ or $\Bbb C$).
A set $\mathrm{A}$ is balanced or circled if for every scalar $\lambda$ with $|\lambda| \leq 1,$ $\lambda \mathrm{A} \subset \mathrm{A}.$
A set $\mathrm{A}$ absorbs another set $\mathrm{S}$ if there exists a number $\alpha > 0$ such that, for whatever scalar $\lambda$ with $|\lambda| < \alpha$ may be, $\lambda \mathrm{S} \subset \mathrm{A}.$
Let $\mathrm{A}$ be a balanced subset of $\mathrm{V}.$ Then, for $\mathrm{A}$ to be a neighbourhood of zero (a set containing an open set for which zero belongs to) it is a necessary and sufficient condition that $\mathrm{A}$ absorbs the range of every sequence converging to zero.
One side is really easy, it is just an immediate consequence that every finite set is bounded (it is absorbed by every neighbourhood of zero; reduce to one point by induction and then use continuity of the mapping $(\lambda, x) \mapsto \lambda x$). For the other, I am not sure if I am heading the right way: assume $\mathrm{A}$ is balanced and absorbs the range of every sequence converging to zero but it does not contain an open set containing zero. Since the space is metrizable, it possesses a denumerable fundamental system of open sets (a family of open sets such that the zero vector belongs to every member of the family and any other open set for which zero belongs to contains a member of the family). Say, $(\mathrm{G}_n)$ is said fundamental system. Without loss of generality, one can assume that said system is decreasing (by, say, choosing an invariant under translations metric on $\mathrm{V}$ and then defining $\mathrm{G}_n$ to be the ball of centre 0 and radius $\frac{1}{n}$; in fact, one can assume more on the $\mathrm{G}_n,$ such as being balanced). Since no $\mathrm{G}_n$ can be a subset of $\mathrm{A},$ there are points $g_n \in \mathrm{G}_n$ not belonging to $\mathrm{A}.$ Then, the sequence $(g_n)$ converges to zero and so, its range is absorbed by $\mathrm{A}.$ This is where I am stuck, no idea what to do next. Any help is welcome.