Inequality in proof of SLLN This comes from theorem 5.1.2 of KL Chung's A Course in Probability Theory. 
Suppose ${X_n}$ are uncorrelated and their second moments have a  common bound.
Then For each $n \ge 1 $, $D_n:= \max_{n^2 \le k < (n+1)^2 } |S_k - S_{n^2}| $, we have 
$$E[|D_n|^2] \le 2n E\left[\left|S_{(n+1)^2}- S_{n^2}\right|^2\right] .$$
This is the only part of the proof I couldn't figure out. I'm sure its some silly trick that I somehow couldn't see. 

Since
\begin{align}
& D_n^2 = \max_{n^2 \leq k < (n + 1)^2 }  \left|S_k - S_{n^2}\right|^2 \\
\leq & \sum_{k = n^2}^{(n + 1)^2 - 1} \left|S_k - S_{n^2}\right|^2 \\
= & \sum_{k = n^2 + 1}^{(n + 1)^2 - 1} \left|S_k - S_{n^2}\right|^2 
\end{align}
It follows that
\begin{align}
& E\left(D_n^2\right) \leq \sum_{k = n^2 + 1}^{(n + 1)^2 - 1}E\left(\left|S_k - S_{n^2}\right|^2\right) \\
\leq & [(n + 1)^2 - 1 - (n^2 + 1) + 1] E\left(\left|S_{(n + 1)^2} - S_{n^2}\right|^2\right) \\
= & 2nE\left(\left|S_{(n + 1)^2} - S_{n^2}\right|^2\right)
\end{align}
where we used for every $k \in \{n^2 + 1, \ldots, (n + 1)^2 - 1\}$, \begin{align}
& E\left(\left|S_k - S_{n^2}\right|^2\right) = E\left[\left(X_{n^2 + 1} + \cdots + X_k\right)^2\right] = \sum_{i = n^2 + 1}^k E\left(X_i^2\right) \\
\leq & \sum_{i = n^2 + 1}^{(n + 1)^2} E\left(X_i^2\right) = E\left(\left|S_{(n + 1)^2} - S_{n^2}\right|^2\right)\end{align}
in which we used uncorrelated condition.
