Let $X_0,X_1,\ldots$ be a sequence of independent random variables such that $\E X_n=0$ and put $S_n\colon=X_0+\cdots+X_n$. If $S_n\to S$ a.s., then $S_n$ is uniformly integrable. Hence $S_n$ converges in $L_1(\P)$ to $S$.
Putting $C\colon=\sup\E|S_n|$ we conclude by Fatou's lemma that:
$$
\E|S|\leq\liminf\E|S_n|\leq C~.
$$
Put $\F_n\colon=\s(X_0,\ldots,X_n)$, then $S-S_n$ is independent from $\F_n$. Hence:
$$
\E(|S-S_n|;|S_n| > t)
=\E|S-S_n|\P(|S_n| > t)
\leq2C\P(|S_n| > t)
$$
and therefore:
\begin{eqnarray*}
\E(|S_n|;|S_n| > t)
&\leq&\E(|S|;|S_n| > t)+\E|S-S_n|\P(|S_n| > t)\\
&\leq&3C\P(|S_n| > t)
\leq3C\P(|S_n-S| > t/2)+3C\P(|S| > t/2)~.
\end{eqnarray*}
Thus for all $\e > 0$, all $t\geq t(\e)$ and all $n\geq n(\e)$:
$$
\E(|S_n|;|S_n| > t)\leq6C\e~.
$$