You have 
$$ 
\begin{aligned}
&\sup_{n\ge1}\sqrt{E[d\left( \mathscr{S}_n^{(v) }, \mathscr{S}_n \right)^2]} \\ 
&=\sup_{n\ge1}\|d\left( \mathscr{S}_n^{(v) }, \mathscr{S}_n \right) \|_2 \leq 2 \sqrt{E \left[ X_1^2; |X_1| > v \right] } \text{ for all } v>0. 
\end{aligned}
$$
So, by the Chebyshev/Markov inequality,
$$ 
\begin{aligned}
&\lim_{v \to \infty} \sup_n P \left( d \left( \mathscr{S}_n^{(v) }, \mathscr{S}_n \right) > \lambda \right) \\ 
&\le\lim_{v \to \infty} \sup_n 
\frac{E[d\left( \mathscr{S}_n^{(v) }, \mathscr{S}_n \right)^2]}{\lambda^2} \\ 
&\le\lim_{v \to \infty}  
\frac{4E \left[ X_1^2; |X_1| > v \right] } 
{\lambda^2} 
=0,
\end{aligned}
$$
because, by the dominated convergence theorem, $E \left[ X_1^2; |X_1| > v \right]\to0$ as $v\to\infty$.