Convergence rate of a matrix of eigenvectors

Question

my problem is the following:

Let ∑₀ be a symmetric matrix and ∑ its estimator. Also define F₀ and its estimator F such that |∑ - ∑₀|=O(T^-1/2) and | F - F₀|= O(N^-1/2). After some manipulations that involved other matrices/processes, I get that |∑ - F₀∑₀F₀^t | = O(T^-1/2 + N^-1/2) (where I use superscript t for transpose) and also |F∑F^t - F₀∑₀F₀^t| = O(T^-1/2 + N^-1/2). Now let S₀ be the matrix of eigenvectors associated with the r distinct positive eigenvalues of F₀∑₀F₀^t an S is the matrix of eigenvectors associated with r distinct positive eigenvalues of ∑.

The convergence rate of |S-S₀| is what?

I believe we have |S-S₀| = O(T^-1/2 + N^-1/2) but I couldn't prove it. I think it is necessary to use the Bauer-Fike Theorem (Theorem 7.2.2 of Golub and Van Loan 2013) but algebra is not my strongest point. Can someone please present-me a comprehensive proof of this.

Thanks.