  The Gallager bound is well known in the area of channel coding. However, most
discussions about it mainly focus on its applications to memoryless channels.
We show in this paper that the bounds obtained by Gallager's method are very
tight even for general sources and channels that are defined in the
information-spectrum theory. Our method is mainly based on the estimations of
error exponents in those bounds, and by these estimations we proved the direct
part of the Slepian-Wolf theorem and channel coding theorem for general sources
and channels.
