Positive lower density of large prime gaps

For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed $C\log p$, where p is the smaller prime. The proportion is bounded below for every sufficiently large initial segment of the prime sequence, with a constant depending on C. It follows that the indices at which $p_n/n$ increases have positive lower asymptotic density, answering a question of Erdős and Prachar.