$F_n$ is a filtration for IID random sequence $X_n$.
I want to prove that when you pull the absolute value to the outside of the Expectation, then it causes an inequality to be introduced. For example:
$E[X_n|F_n] = E[X_n|F_n]$
Take absolute value and expectation of both side.
$E\Big[~\Big|E[X_n|F_n]\Big|~\Big] = E\Big[~\Big|E[X_n|F_n]\Big|~\Big]$
on the RHS, Pull the absolute value to the outside of the expectation:
$E\Big[~\Big|E[X_n|F_n]\Big|~\Big] \le \Bigg|E\Big[~E[X | F_n \Big]~ \Bigg|$
Here an inequality is introduced. Why is this true?
