Let $X_1,X_2,..$ are i.i.d. random variables with $E(|X_1|)<\infty$. Define the partial sum $S_n = X_1+X_2+...+X_n$.
Show that $Y_n=S_n-n \mu$ is a martingale where $E(X_1)=\mu$.
I took $\mathcal{F}_n= \sigma(X_1,..X_n)$, and I proved that $E(Y_{n+1}|\mathcal{F_n})=Y_n$ and $Y_n \in \mathcal{F_n}$.
Now, I want to show that $E(|Y_n|)<\infty$.
$E(|Y_n|)=E(|S_n - n \mu|)\leq E(|S_n|)+n \mu$.
How do I say the right hand side term of the above equation is finite?