Yes, sorry it was my mistake i read inequality wrong. I'm deleting my last comment.

Okay, I'll make a note of it. Thank you very, very much.

I'm so sory but i don't know one thing to understand entire proof. How doeas the the dominated convergence theorem is used here?

There should be $\sup_{n\ge1}\sqrt{E[d\left( \mathscr{S}_n^{(v) }, \mathscr{S}_n \right)^2]}$ right? But i have another stupid question. Why the $\sup_{n\ge1}\sqrt{E[d\left( \mathscr{S}_n^{(v) }, \mathscr{S}_n \right)^2]} = \sup_{n\ge1}||{d\left( \mathscr{S}_n^{(v) }, \mathscr{S}_n \right)}||_2$ ?