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Insert one factor difference at a time: the $j$th term is $U_m\cdots U_{j+1}(U_j-V_j)V_{j-1}\cdots V_1$. Multiplication by the surrounding unitary operators preserves the spectral norm, so the triangle inequality gives the displayed dimension-independent bound. No commutation assumption is required.

A universal quantum gate set can approximate every finite-qubit unitary operator arbitrarily accurately using finite circuits over the set.

For a unitary operator $U$ on a Hilbert space, the Cesaro averagesconverge strongly to the orthogonal projection onto the fixed-point subspace $\ker (U-I)$.