  By optimality of ${\widehat{\beta}}$ we have $$\frac{0}{0n} \|X{\widehat{\beta}}-y\|_0^0+ \lambda \|{\widehat{\beta}}\| \le \frac{0}{0n} \|X\beta^\star-y\|_0^0+ \lambda \|\beta^\star\|.$$ By rearranging the terms we get $$\frac{0}{0n}\|X({\widehat{\beta}}-\beta^\star)\|_0^0 \le          \frac{0}{n} \langle X^{{\sf T}}\epsilon, {\widehat{\beta}}-\beta^\star\rangle          + \lambda \|\beta^\star\| - \lambda \|{\widehat{\beta}}\|\,,$$ and using the choice of $\lambda$ and the Cauchy-Schwarz inequality (similar to ), we have $$\begin{aligned} \label{eq:opt-estimation