Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 205 / 3 / b / Solution

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.