varying degrees of modifications from similar lemmas in [@Zhang09-multistage], and thus the proofs are included for completeness.

\[lem:sub-Gaussian-infnorm\] Define $\hat{\epsilon}= \frac{1}{n} X^\top (X \tilde{{{\mathbf w}}}-{{\mathbf y}})$. Under the conditions of Assumption \[assump:fixed\], with probability larger than $1-\eta$: $$\forall j \in \bar{F}: \quad |\hat{\epsilon}_j|=0 , \quad
  |\tilde{{{\mathbf w}}}_j- \bar{{{\mathbf w}}}_j| \leq \sigma \sqrt{2 \rho_-(\bar{k})^{-1} \ln (2p/\eta)/ n}  ,$$ and $$\forall j \notin \bar{F}: \quad |\hat{\epsilon}_j|\leq \sigma \sqrt{2 \rho_+(1) \ln (2p/\eta)/ n}  .$$

Let $\tilde{P}$ be the projection matrix to the subspace spanned by columns of $X$ in $\bar{F}$, then we know that $$X \tilde{{{\mathbf w}}} = \tilde{P} {{\mathbf y}}$$ and $$(I-\tilde{P}) {{\mathbf E}}{{\mathbf y}}= {{\mathbf E}}{{\mathbf y}}-  X \bar{{{\mathbf w}}} = 0 .$$ Therefore for each $j$ $$n |\hat{\epsilon}_j| =  |X_j^\top (X \tilde{{{\mathbf w}}}-{{\mathbf y}}) | = 
|X_j^\top (I-\tilde{P}) ({{\mathbf y}}- {{\mathbf E}}{{\mathbf y}})) | .$$ It implies that $\hat{\epsilon}_j=0$ if $j \in \bar{F}$. Since for each $j$: the column $X_j$ satisfies $\|X_j^\top (I-\tilde{P})\|_2^2 \leq n \rho_+(1)$, we have from sub-Gaussian tail bound that for all $j \notin \bar{F}$ and $\epsilon>0$: $$P \left[ |\hat{\epsilon}_j| \geq \epsilon \right] \leq 2 \exp [ - n\epsilon^2/(2 \sigma^2 \rho_+(1)) ] .$$ Moreover, for each $j \in \bar{F}$, we have $$|\tilde{{{\mathbf w}}}_j-\bar{{{\mathbf w}}}_j|= e_j^\top (X_{\bar{F}}^\top X_{\bar{F}})^{-1} 