Prove that this sequence is a submartingale Let $X_1, X_2,\cdots$ be i.i.d. random variable on $(\Omega, \mathbb F)$ with $EX_1=0$ and $VX_1 = \sigma^2$. Consider a filtration $\mathbb F_n = \mathbb F(X_1,\cdots, X_n)$.

I want to show that $Y_n = \left( \sum_{k=1}^n X_k \right)^2$ is a
  submartingale.

This is my attempt:
\begin{eqnarray}
E[Y_{n+1} | \mathbb F_n] &= E \left[ \left( \sum_{k=1}^{n+1} X_k \right)^2 \,\,\big|\,\, \mathbb F_n \right] \\
&= E \left[ \sum_{i,j=1}^{n+1} X_iX_j \,\,\big|\,\, \mathbb F_n \right] \\
&= \sum_{i,j=1}^{n+1} E \left[ X_i \,\,\big|\,\, \mathbb F_n \right] E \left[ X_j \,\,\big|\,\, \mathbb F_n \right] \\
&= \sum_{i,j=1}^{n} X_i X_j + 2 \sum_{i=1}^{n+1} X_i E \left[ X_{n+1} \,\,\big|\,\, \mathbb F_n \right] \\
&= Y_n + 2 \sum_{i=1}^{n+1} X_i E \left[ X_{n+1} \,\,\big|\,\, \mathbb F_n \right]
\end{eqnarray}
How can I proceed from here? Of course if I knew $X_i\geq 0$ I would be done, but as is I don't know what to do next.
Moreover, I'm not sure how to show that $E |Y_n| < \infty$ when we don't know that $X_i\geq 0$.
Any help is MUCH appreciated.
 A: $\newcommand{\E}{\mathbb E}\newcommand{\F}{\mathcal F}\newcommand{\Var}{\operatorname{Var}}$First let's do the integrability part of $Y_n$:
\begin{align}
\E|Y_n|=\E\left[\left(\sum_{k=1}^n X_k\right)^2\right]=\Var\left(\sum_{k=1}^n X_k \right)+\left(\E\left[ \sum_{k=1}^n X_k \right]\right)^2 <\infty
\end{align}
Where we have used $\E[Z^2]=\Var(Z)+\E[Z]^2$. Now let's show the rest: \begin{align}
\E[Y_{n+1}|\F_n]&=\E\left[\left(\sum_{k=1}^{n+1} X_k\right)^2\  \Bigg |\  \F_n\right]\\
&=\E\left[\sum_{k=1}^{n+1}X_k^2+2\sum_{j=1}^{n+1}\sum_{k=1}^{j-1}X_kX_j \ \Bigg |\ \F_n \right]\\
\end{align}
where the formula in this post is used. By linearity we have:
\begin{align} 
\E[Y_{n+1}|\F_n]&=\sum_{k=1}^n X_k^2+\E[X_{n+1}^2]+2\sum_{j=1}^{n+1}\sum_{k=1}^{j-1}X_k\E[X_j|\F_n]\\
&=\sum_{k=1}^n X_k^2+\E[X_{n+1}^2]+2\sum_{j=1}^{n}\sum_{k=1}^{j-1}X_k\E[X_j|\F_n]+2\sum_{k=1}^{n}X_k\E[X_{n+1}|\F_n]\\
\end{align}
where we have used the pull-out property of "known" variables and we will use it once again. The most right term is $\E[X_{n+1}|\F_n]=\E[X_{n+1}]=0$ by assumption hence:
\begin{align}
\E[Y_{n+1}|\F_n]&=\sum_{k=1}^n X_k^2+\E[X_{n+1}^2]+2\sum_{j=1}^{n}\sum_{k=1}^{j-1}X_kX_j\\
&\geq \sum_{k=1}^n X_k^2+2\sum_{j=1}^{n}\sum_{k=1}^{j-1}X_kX_j\\
&= \left(\sum_{k=1}^n X_k \right)^2\\
&= Y_n
\end{align}
So $(Y_n,\F_n)$ is a submartingale.
