Unitary product telescoping ($\Vert U_m\cdots U_1-V_m\cdots V_1\Vert \le {\sum }_j\Vert U_j-V_j\Vert$)

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.