Question: Given a matrix $X \in \mathbb{R}^{m \times n}$. Let $(X^k)_k \in \mathbb{R}^{m \times n}$ is a sequence converge to $X$. Denote $\sigma_i(X)$ as singular values of $X$. Prove that $$\lim_{k \to \infty} \sigma_i(X^k) = \sigma(X).$$
I wonder that my question is feasible or not?
