Unitary product telescoping
= Unitary product telescoping
{title2=$\|U_m\cdots U_1-V_m\cdots V_1\|\leq\sum_j\|U_j-V_j\|$}
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.