  In \cite{RH3} Rasin and Hydon suggested a way to construct an infinite number
of conservation laws for the discrete KdV equation (dKdV), by repeated
application of a certain symmetry to a known conservation law. It was not
decided, however, whether the resulting conservation laws were distinct and
nontrivial. In this paper we obtain the following results: (1) We give an
alternative method to construct an infinite number of conservation laws using a
discrete version of the Gardner transformation. (2) We give a direct proof that
the Rasin-Hydon conservation laws are indeed distinct and nontrivial. (3) We
consider a continuum limit in which the dKdV equation becomes a first-order
eikonal equation. In this limit the two sets of conservation laws become the
same, and are evidently distinct and nontrivial. This proves the nontriviality
of the conservation laws constructed by the Gardner method, and gives an
alternate proof of the nontriviality of the conservation laws constructed by
the Rasin-Hydon method.
