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The question is:

5.2.11. Let $X_n$ and $Y_n$ be positive integrable and adapted to Fn. Suppose $\mathbb E(X_{n+1}|\mathcal F_n) ≤ (1 + Y_n )X_n$ with $ \sum Y_n < \infty$ a.s. Prove that $X_n$ converges a.s. to a finite limit by finding a closely related supermartingale.

My guess is that we can define $Z_n = \prod_{m \le n} (1+Y_m)$. Then the given inequality can be rewritten as $$ \mathbb E(X_{n+1}/Z_n|\mathcal F_n) ≤ X_n /Z_{n-1}. $$ Let $S_n = X_n / Z_{n-1}$. As long as $S_n$ is integrable for all $n$ (I think they are because $Z_n \ge 1$), $S_n$ is a positive super martingale. So by martingale convergence theorem, we have $X_n / Z_{n-1}$ converges almost surely to a finite limit. By this question, we have $Z_n$ converges almost surely. It follows that $X_n$ converges almost surely.

So is this a valid proof?

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Looks good. I remember this exercise; the hardest part was finding the recipe for the supermartingale. – A Blumenthal Feb 21 '13 at 20:37

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