  We prove that the arithmetic degree of a graded or local ring is bounded
above by the arithmetic degree of any of its associated graded rings with
respect to ideals $I$ in $A$. In particular, if $Spec (A)$ is equidimensional
and has an embedded component in dimension $i$, then the normal cone of $Spec
(A)$ along $V(I)$ has an embedded component in dimension $i$ too.
