Let $X_0,X_1,\dots$ be integrable and non-negative random variables. Let $\mathcal{F}_n:=\sigma(X_0,\dots,X_n)$ and suppose that $\mathbb{E}(X_{n+1}\mid\mathcal{F}_n)\leq b_n + X_n$ where $(b_n)$ is a sequence of non-negative real numbers with $\sum\limits_{n=0}^\infty b_n<\infty$.
Show that $X_\infty:=\lim\limits_{n\to \infty}X_n$ exists almost surely and $X_\infty$ is integrable.
(Hint: Consider the $Y_n=X_n+\sum\limits_{k=n}^\infty b_k$)
So if I can show that $X_n$ is a sub- or supermartingale and that $\sup E(|X_n|)<\infty$ then I know that $X_\infty:=\lim\limits_{n\to \infty}X_n$ exists almost surely and $X_\infty$ is integrable. Unfortunately I don't know how to show that it's a sub- or a supermartingale since the $b_n$ are unkown.
I tried to work with the hint $Y_n=X_n+\sum\limits_{k=n}^\infty b_k$ $$E(Y_{n+1}\mid \mathcal{F}_n)=E(X_{n+1}+\sum\limits_{k=n+1}^\infty b_k\mid \mathcal{F}_n)=E(X_{n+1}\mid \mathcal{F}_n)+\sum\limits_{k=n+1}^\infty b_k$$
But I don't see how this is useful.
Since $Y_n \ge 0$, if you can show that $Y_n$ is a supermartingale, the you are done. In fact, \begin{align*} E(Y_{n+1}\mid \mathcal{F}_n) &= E(X_{n+1}\mid \mathcal{F}_n) + \sum_{k=n+1}^\infty b_k\\ &\le X_n + b_n + \sum_{k=n+1}^\infty b_k= Y_n. \end{align*} Then $\lim_{n\rightarrow\infty} Y_n$ exists, and, consequently, $\lim_{n\rightarrow\infty} X_n$ exists.