In the the paper "Asymptotics for Lasso-Type Estimators" by K.Knight and W.J. Fu, Ann.Statist. At the end of Theorem 1, the auther said that $$Z_n(\Phi)\geq\frac{1}{n}\sum_{i=1}^{n}(Y_i-x_i^T\Phi)^2=Zn^{(0)}(\phi).$$ for all $\phi$. Since argmin($Z_n^{(0)}$)=$O_p(1)$, it follows that argmin($Z_n$)=$O_p(1)$.
Is there anybody can give me some hint or suggest some reference on why the last sentence holds: Since argmin($Z_n^{(0)}$)=$O_p(1)$, it follows that argmin($Z_n$)=$O_p(1)$
I think I have made it. Here is the main idea. Since $Z_n(\Phi)=Z_n^{(0)}(\Phi)+\lambda_n/n\sum|\phi_i|^\gamma$, and the part on the righthand side of plus is always nonnegative, the minimizer of $Z_n^{(0)}(\Phi)$ should also minimize $Z_n(\Phi)$. Ortherwise, the contradiction can be easily deduced.