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 $X$)?



Limitation through the singular values
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.


