This is an exercise mentioned in this question:
Let $Y$ be an integrable random variable on the space $(\Omega,{\mathcal A},{\bf P})$ and $\mathcal{G}$ be a sub $\sigma$-algebra of $\mathcal{A}$. Show that $|Y|\leq c$ implies $|E[Y\mid{\mathcal G}]|\leq c$.
It has been shown that using $E[|y|\mid{\mathcal G}]$ to estimate $|E[Y \mid{\mathcal G}]|$ is not a good idea due to did's answer to the previous question.
One more step I can go so far is $E[Y]=E[E[Y\mid{\mathcal G}]]$. I don't think this would work since expectation only gives the average.
So how can I get $|E[Y\mid{\mathcal G}]|\leq c$ using $|Y|\leq c$?