Single-qubit gates and CNOT are universal

Let $n\ge1$. The subgroup of $U(2^n)$ generated by all single-qubit gates $U^{(k)}$ ($U\in U(2)$, $k$ a qubit) together with all CNOT gates $\mathrm{CNOT}^{(i,j)}$ on ordered pairs of distinct qubits is the whole unitary group:

Every $n$-qubit unitary is therefore an exact finite product of single-qubit gates and CNOTs; this is the classical universality theorem of the gate model, here used as the stepping stone to the iSWAP version.

Formalization Note For $n=1$ there are no qubit pairs and the statement reduces to $U(2)$ being generated by single-qubit gates; $n=0$ is excluded because then no generators exist while $U(1)$ is nontrivial.