Linear finite-dimensional topological vector space is closed.

Question

Suppose $X$ is a topological vector space and let $Y \subset X$ be its subspace with $\dim Y < \infty$. The goal is to prove that $Y$ is closed in $X$. I know how to prove this fact when $X$ is endowed with a norm by picking up a Cauchy sequence, but could you please suggest me, how to do it when we do not have any norm?