Limitation through the singular values

Question

Given matrix $X \in \mathbb{R}^{m\times n}$ and sequence $\left\{X^k\right\}_k$ converges to $X$ according to the Frobenius norm. I wonder that $\sigma_i(X^k)$ converge $\sigma_i(X)$ or not (where $\sigma_i(X)$ is singular values of $X$)?

Answer 1

Yes, one can prove that $$ |\sigma_i(A+E) - \sigma_i(A)| \leq \|E\| \quad \quad \forall i $$ which implies that the singular values are continuous. This follows, for instance, from Weyl's inequalities.

See also https://math.stackexchange.com/q/2783345/65548 , which links to a proof of these inequalities by Qiaochu Yuan.