Suppose I have a sequence of random variables $\{X_n\}_{n=1}^{\infty}$ taking values in $\mathbb{R}^{k\times k}$ and $S \subset \mathbb{R}^{k\times k}$ is a given bounded open set. If I know that$$\mathbb{E}[|X_n|^2] < \frac{c_1}{n}$$where $|\cdot|$ is the spectral norm on vector space of matrices, then how can I prove or disprove that $\lim_{n \to \infty}\mathbb{P}(X_n \in S') \to 0$ (where $S'$ denotes the complement of $S$)?
For reference, the spectral norm is defined here: https://mathworld.wolfram.com/SpectralNorm.html .
