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Let $\{X_n : n \geq 1\}$ be an independent sequence of nonnegative random variables and :

$S_n= X_1 +...+X_n$.

Show that $\{S_n\}$ is a submartingale.

This is what I have so far: I know it has to follow that $E[X_{n+1}|X_1,...,X_n] \geq X_n.$ Since they are nonnegative then I have to show that $E[X_n^+|X_1,...X_n^+] \geq S_n$ if this is the wrong aproach then how do I tackle this?

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Linearity of conditional expectation gives $E(S_{n+1}|\mathcal{F}_n)=\sum_{k=1}^{n+1} E(X_k|\mathcal{F}_n)$. Since $X_k\in\mathcal{F}_n$ for $k\le n$ (the values of each of these random variables is known, if you know $\mathcal{F}_n$), we have $E(X_k|\mathcal{F}_n)=X_k$. So we have $$E(S_{n+1}|\mathcal F_n)=\sum_{k=1}^n X_k +E(X_{n+1}|\mathcal F_n)=S_n+E(X_{n+1}|\mathcal F_n)\ge S_n,$$ with inequality since the conditional expectation of a nonnegativity random variable is nonnegative.

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