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We provide a general framework to study stochastic sequences related to individual learning in economics, learning automata in computer sciences, social learning in marketing, and other applications. More precisely, we study the asymptotic properties of a class of stochastic sequences that take values in $[0,1]$ and satisfy a property called "bounded expected relative hazard rates." Sequences that satisfy this property and feature "small step-size" or "shrinking step-size" converge to 1 with high probability or almost surely, respectively. These convergence results yield conditions for the learning models in B\"orgers, Morales, and Sarin (2004), Erev and Roth (1998), and Schlag (1998) to choose expected payoff maximizing actions with probability one in the long run., Comment: After revision. Accepted for publication by Journal of Economic Theory