Let $(X_n)_{n\geq 1}$ be a sequence of independent random variables in $L^2$. Let $S_n = X_1 + \cdots + x_n$ and write $$\mu_n = \mathbb{E}[S_n], \sigma_n^2 = var(S_n)$$ Show that b) implies a) where,
a) the sequences $\mu_n$ and $\sigma_n^2$ converge in $\mathbb{R}$
b) there exists a RV S such that $S_n \rightarrow S$ a.s. and in $L^2$
I am struggling to show that $\sigma_n^2$ converges.
We have that $\sigma_n^2 = \mathbb{E}[S_n^2] - \mathbb{E}[S_n]^2$. Since $S_n \to S$ in $L^2$ we also have that $\mathbb{E}[S_n^2] \to \mathbb{E}[S^2]$ as $n \to \infty$. Additionally, $\mathbb{E}[S_n] \to \mathbb{E}[S]$ since by Holder's inequality, $|\mathbb{E}[S_n - S]| \leq \|S_n- S\|_{L^2}$. In particular, $\sigma_n^2 \to \mathbb{E}[S^2] - \mathbb{E}[S_n]^2 = \operatorname{Var}(S)$.