Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 1 / d / Solution

The strong law of large numbers givesBecause $0<\lambda <1$, it follows that ${\lambda }^{S_n}\to 0$ almost surely. If this martingale were uniformly integrable, almost-sure convergence would imply convergence in $L^1$, and thereforeBut the martingale has constant expectation $\mathbb{E}[{\lambda }^{S_n}]=1$. This contradiction proves that