It was shown in Flesch and Solan (2022) with a rather involved proof that all two-player stochastic games with finite state and action spaces and shift-invariant payoffs admit an $ε$-equilibrium, for every $ε>0$. Their proof also holds for two-player absorbing games with tail-measurable payoffs. In this paper we provide a simpler proof for the existence of $ε$-equilibrium in two-player absorbing games with tail-measurable payoffs, by combining recent mathematical tools for such payoff functions with classical tools for absorbing games.