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A compatibility condition lower-bounds prediction norm by a scaled ${\ell }^1$ norm on a Lasso cone condition. One convention requires $|S|\Vert X\delta {\Vert }_2^2/n\ge {\varphi }^2\Vert {\delta }_S{\Vert }_1^2$ throughout the cone.

Write $\delta =\hat{\beta }-{\beta }^0$. On $\Omega$, Hölder's inequality givesSince ${\beta }_N^0=0$, the triangle inequality givesSubstitution in the Basic inequality for the Lasso, followed by discarding the nonnegative prediction-error term, yieldsTherefore $\Vert {\hat{\beta }}_N-{\beta }_N^0{\Vert }_1\le 3\Vert {\hat{\beta }}_S-{\beta }_S^0{\Vert }_1$, the Lasso cone condition.

Part b places $\delta =\hat{\beta }-{\beta }^0$ in the Lasso cone condition. Keeping the prediction-error term in the same argument giveswhere the second step is the Cauchy-Schwarz inequality. The restricted eigenvalue condition givesfor nonzero $\delta$ in this cone. Division by $\Vert \delta {\Vert }_2$ provesThe result is immediate when $\delta =0$.