content_image/309559.jpg content_image/309558.jpg content_image/309557.jpg content_image/309556.jpg content_image/309555.jpg
to Remark 6, we can consider in R p × q × R q × q the topology induced by the norm 

<img src='content_image/309559.jpg'>

Given ( B$_{0}$ , Σ$_{0}$ ) in R p × q × R q × $^{q}$, the proof consists in find an upper bound of | d $^{2}$( B , Σ ) − d $^{2}$( B$_{0}$ , Σ$_{0}$ ) | that tends to 0 when ‖| ( B , Σ ) − ( B$_{0}$ , Σ$_{0}$ ) ‖| → 0 . 

Adding and subtracting d $^{2}$( B$_{0}$ , Σ ) we have that 

<img src='content_image/309558.jpg'>

Let r ( B$_{0}$ ) = ( y − B ′ $_{0}$x ). Using basic tools from linear algebra we obtain 

<img src='content_image/309557.jpg'>

and by Weyl’s Perturbation Theorem (see [4], pg. 63), we have that 

<img src='content_image/309556.jpg'>

combining these inequalities we obtain a bound of | d $^{2}$( B , Σ ) − d $^{2}$( B$_{0}$ , Σ ) | that tends to 0 when ‖| ( B , Σ ) − ( B$_{0}$ , Σ$_{0}$ ) ‖| → 0 . 

If r ( B$_{0}$ ) = 0 the lemma is proved, otherwise using the Cauchy-Schwarz in- equality and (A.6) we have 

<img src='content_image/309555.jpg'>

which completes the proof. ■ 

Proof of Theorem 1: By Lemma 9, it sufices to show that there exist t$_{1}$ and t$_{2}$ 

