On upper topological limit of family of vector subspaces of codimension $k$

Let $\{L_\alpha\mid \alpha\in I\}$ be an infinite family of subspaces in a topological vector space $X$ the codimension of each of which is at most $k$. We prove that there exists a subspace $L\subset X$, $\operatorname{codim} L\leq k$, such that every $x\in L$ is a limit point of some family $\{l_\alpha\in L_\alpha\}$.