Solution
= 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, so
$$
L_{Z'}(0)\leq h\rho(M_y).
$$
The $L^1$ regularization term vanishes at zero.