I’m looking for a correct technical version (and in the best case a reference) for a statement of the following type:

Consider a complex algebraic variety $X\subset\mathbb{P}^n$ and a sequence of linear spaces $(L_n)_{n\in\mathbb{N}}$ that is convergent in the Grassmannian $\mathbb{G}(k+1,n+1)$ of k-dimensional linear subspaces of $\mathbb{P}^n$ (in the euclidean topology). Let $L\in\mathbb{G}(k+1,n+1)$ be the limit of this sequence. Now suppose $L_j\cap X$ is 0-dimensional for every $j\in\mathbb{N}$ and the length of the scheme $L_j\cap X$ is equal to m for every $j\in\mathbb{N}$. Then I would like to conclude that the length of $L\cap X$ is also m, given that $L\cap X$ is also 0-dimensional.