  The aim of this paper is to provide a self-contained proof of a general case
of the coarea inequality, also known as the Eilenberg inequality. The result is
known, but we are not aware of any place that a proof would be written with all
details. The known proof is based on a difficult result of Davies. Our proof is
elementary and does not use Davies' theorem. Instead we use an elegant argument
that we learned from Nazarov through MathOverflow. We also obtain some
generalizations of the coarea inequality.
