I am looking at a certain proof of asymptotic normality of M-estimators and there is a matrix $V$ which is known to be invertible anda random variable $X_n = V + o_p(1)$ (that is, the difference converges to zero in probability as $n \to \infty$). I want to take the inverse of X_n here. The proof goes that $X_n^{-1} = V^{-1} + o_p(1)$ by a continuous mapping theorem for convergence in probability.
This step is fine, but my question is how we can justify the fact that X_n can be invertible for large n. I know that if a matrix is close enough to an invertible matrix it will be invertible itself because the determinant is a continuous function, but I am unable, to come up with an extension of the continuous mapping theorem that allows for X to be in the domain only with arbitrarily high probability (but not Probability 1). I am not sure if that is enough for the rest of the proof.
Here is an image of th statement and proof and my issue is at the end.
