  The notion of almost everywhere convergence has been generalized to vector
lattices as unbounded order convergence, which proves a very useful tool in the
theory of vector and Banach lattices. In this short note, we establish some new
results on unbounded order convergence that tie up some loose ends. In
particular, we show that every norm bounded positive increasing net in an order
continuous Banach lattice is uo-Cauchy and that every uo-Cauchy net in an order
continuous Banach lattice has a uo-limit in the universal completion.
