The lasso error vector lies in a cone

Let $\hat\beta$ minimise $\frac{1}{2n}\|y-X\beta\|_2^2 + \lambda\|\beta\|_1$ with $y = X\beta^\star+w$, and let $S = \operatorname{supp}(\beta^\star)$, $v = \hat\beta-\beta^\star$. Prove that on the event $\frac1n\|X^\top w\|_\infty \le \lambda/2$,

$$\|v_{S^c}\|_1 \le 3\|v_S\|_1 .$$

Explain why this is the step that makes a restricted eigenvalue condition — rather than a full eigenvalue condition — sufficient.