If all the vector spaces in a convergent sequence in the Grassmanian contains a vector, then the limit of the sequence contains the vector?

Question

Let $G(k,n)$ denote the Grassmanian of $k$-dimensional vector subspaces in $\mathbb R^n$. Suppose $W_n$, $n \in \mathbb N$ is a sequence of subspaces in $G(k,n)$ converging to some $W$ in its topology. If there is $0 \neq v$ with $v \in W_n$ for all $n \in \mathbb N$, then is it true that $v \in W$ also? How would one prove this if true?