  We present a new proof of the compactness of bilinear paraproducts with CMO
symbols. By drawing an analogy to compact linear operators, we first explore
further properties of compact bilinear operators on Banach spaces and present
examples. We then prove compactness of bilinear paraproducts with CMO symbols
by combining one of the properties of compact bilinear operators thus obtained
with vanishing Carleson measure estimates and interpolation of bilinear
compactness.
