# Why are these the conditions for convergence of reward values?

In Reinforcement Learning: An Introduction by Barto and Sutton, there's the statement that by using the update rule $$Q_{n+1} \leftarrow Q_n + \alpha_n(R_n - Q_n)$$ that we only have convergence of $$\{Q_n\}$$ almost surely if $$\sum_{i=1}^\infty \alpha_i= \infty$$ and $$\sum_{i=1}^\infty \alpha_i^2< \infty$$, citing this as a "well-known result in stochastic approximation theory." What is this result, and why is is true? My understanding is that we can expand $$Q_{n+1} = \alpha_nR_n + (1-\alpha_n)\alpha_{n-1}R_{n-1} + (1-\alpha_n)(1-\alpha_{n-1})\alpha_{n-2}R_{n-2} + \cdots$$ but I'm not sure how to proceed from here. Do we need to assume that the $$R_i$$ are i.i.d.?