Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 223 / 2 / a / Solution

Changing at most $n-h$ observations leaves at least $h$ original observations unchanged. At $\theta =0$, each unchanged observation has residual $y_i$ and loss at most $\rho (M_y)$. The sum of the $h$ smallest losses cannot exceed the sum over any chosen $h$ observations, soThe $L^1$ regularization term vanishes at zero.