  We prove $\ell^p\big(\mathbb Z^d\big)$ bounds for $p\in(1, \infty)$, of
$r$-variations $r\in(2, \infty)$, for discrete averaging operators and
truncated singular integrals of Radon type. We shall present a new powerful
method which allows us to deal with these operators in a unified way and obtain
the range of parameters of $p$ and $r$ which coincide with the ranges of their
continuous counterparts.
