Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics II / 2 / b / ii / Solution

False. Take the positive geometric sequence $\boxed{a_n=2^{-n}}$. It tends to zero, whileA positive limit of a sequence would make the claim true since $a_{n+1}/a_n$ would then tend to $\ell /\ell =1$, but positivity of each individual term does not exclude a zero limit of a sequence.