  If $A$ and $B$ are $n$- and $m$-representation finite $k$-algebras, then
their tensor product $\Lambda = A\otimes_k B$ is not in general
$(n+m)$-representation finite. However, we prove that if $A$ and $B$ are
acyclic and satisfy the weaker assumption of $n$- and $m$-completeness, then
$\Lambda$ is $(n+m)$-complete. This mirrors the fact that taking higher
Auslander algebra does not preserve $d$-representation finiteness in general,
but it does preserve $d$-completeness. As a corollary, we get the necessary
condition for $\Lambda$ to be $(n+m)$-representation finite which was found by
Herschend and Iyama by using a certain twisted fractionally Calabi-Yau
property.
