  We show that for every $N\ge 3$ the free unitary group $U^+_N$ is
topologically generated by its classical counterpart $U_N$ and the lower-rank
$U^+_{N-1}$. This allows for a uniform inductive proof that a number of
finiteness properties, known to hold for all $N\ne 3$, also hold at $N=3$.
Specifically, all discrete quantum duals $\widehat{U^+_N}$ and
$\widehat{O^+_N}$ are residually finite, and hence also have the Kirchberg
factorization property and are hyperlinear. As another consequence, $U^+_N$ are
topologically generated by $U_N$ and their maximal tori
$\widehat{\mathbb{Z}^{*N}}$ (dual to the free groups on $N$ generators) and
similarly, $O^+_N$ are topologically generated by $O_N$ and their tori
$\widehat{\mathbb{Z}_2^{*N}}$.
