  Let $1\leq p < \infty$ and $1\leq r \leq p^\ast$, where $p^\ast$ is the
conjugate index of $p$. We prove an omnibus theorem, which provides numerous
equivalences for a sequence $(x_n)$ in a Banach space $X$ to be a $(p,r)$-null
sequence. One of them is that $(x_n)$ is $(p,r)$-null if and only if $(x_n)$ is
null and relatively $(p,r)$-compact. This equivalence is known in the "limit"
case when $r=p^\ast$, the case of the $p$-null sequence and $p$-compactness.
Our approach is more direct and easier than those applied for the proof of the
latter result. We apply it also to characterize the unconditional and weak
versions of $(p,r)$-null sequences.
