Let $X_1,X_2,\dots$ be iid integrable random variables in $\mathbb{R}$ with $\mathbb{E}(X_i)=0$. Let $\eta$ be a stopping time with $\mathbb{E}(\eta)<\infty$. Show that $X_\eta$ is integrable.
I'm not sure where to start with this one. Of course, integrability of $\eta$ implies that $\mathbb{P}(\eta<\infty)$ almost surely, and so $X_\eta$ is certainly well-defined. Unfortunately, my only ideas involve thinking about the variable $X_{n\wedge\eta}$ and breaking this down using indicator functions to show that it is integrable, but I can't find a way to show that the fact that this is integrable for any $n$ implies that $X_\eta$ is integrable. I imagine my approach is not the correct one. Any advice would be greatly appreciated!
Set $W = \sum_{k=1}^\eta |X_k| = \sum_{k=1}^\infty |X_k|\mathbf{1}_{\{\eta\geq k\}}$. In order to show that $X_\eta$ is integrable, it is enough to show that $\mathbb{E}[W]< \infty$.
Notice that $$ \mathbb{E}\left[|X_k|\mathbf{1}_{\{\eta\geq k\}}\middle| \mathcal{F}_{k-1}\right] = \mathbf{1}_{\{\eta\geq k\}} \mathbb{E}[|X_k| | \mathcal{F}_{k-1}] = \mathbf{1}_{\{\eta\geq k\}} \mathbb{E}[|X_1|],$$ where I used that $\{\eta\geq k\}$ is $\mathcal{F}_{k-1}$-measurable for the first equality and that the $X_n$ are iid for the second. Thus taking expectations, we have $$\mathbb{E}[|X_k|\mathbf{1}_{\{\eta\geq k\}}] =c \, \mathbb{P}(\eta\geq k).$$ This yields $$\mathbb{E}[W] = \sum_{k=1}^\infty \mathbb{E}[|X_k|\mathbf{1}_{\{\eta\geq k\}}]=c \sum_{k=1}^\infty \mathbb{P}(\eta \geq k) = c \,\mathbb{E}[\eta]<\infty.$$
P.S: This answer is based on a sufficient condition to apply the optional stopping theorem, see condition (b) here.