[proofplan] The proof counts the number of coordinates whose magnitude exceeds the threshold $\tau$. Each such coordinate contributes strictly more than $\tau^q$ to the $\ell_q^q$ sum. Summing these lower bounds over the effective support and using the defining bound $\sum_{j=1}^p |\beta_j^*|^q \leq R$ gives the estimate after rearranging. [/proofplan]

[step:Convert membership in the effective support into a coordinatewise lower bound]Fix $\tau > 0$. For every index $j \in S_\tau(\beta^*)$, the definition of $S_\tau(\beta^*)$ gives $|\beta_j^*| > \tau$. Since $q > 0$, the map $t \mapsto t^q$ from $[0,\infty)$ to $[0,\infty)$ is strictly increasing, hence \begin{align*} |\beta_j^*|^q > \tau^q \end{align*} for every $j \in S_\tau(\beta^*)$.[/step]

[guided]Fix a threshold $\tau > 0$. The effective support $S_\tau(\beta^*)$ consists exactly of those indices whose coordinates are larger than $\tau$ in absolute value: \begin{align*} S_\tau(\beta^*) = \left\{ j \in \{1, \dots, p\} : |\beta_j^*| > \tau \right\}. \end{align*} Thus, if $j \in S_\tau(\beta^*)$, then $|\beta_j^*| > \tau$. We want to compare this pointwise inequality with the $\ell_q^q$ quantity appearing in the definition of $\mathcal{B}_q(R)$. Since $q > 0$, define the function $\varphi: [0,\infty) \to [0,\infty)$ by $\varphi(t) = t^q$ for every $t \in [0,\infty)$. This function is strictly increasing. Applying $\varphi$ to the inequality $|\beta_j^*| > \tau$ preserves the strict inequality and gives \begin{align*} |\beta_j^*|^q > \tau^q \end{align*} for every $j \in S_\tau(\beta^*)$.[/guided]

[step:Sum the coordinatewise lower bound over the effective support] Because $S_\tau(\beta^*)$ is a finite subset of $\{1,\dots,p\}$, the finite sum over $S_\tau(\beta^*)$ is well-defined. If $S_\tau(\beta^*) = \varnothing$, then both $\sum_{j \in S_\tau(\beta^*)} |\beta_j^*|^q$ and $|S_\tau(\beta^*)|\tau^q$ are equal to $0$. If $S_\tau(\beta^*) \neq \varnothing$, summing the strict inequality from the previous step over all $j \in S_\tau(\beta^*)$ gives \begin{align*} \sum_{j \in S_\tau(\beta^*)} |\beta_j^*|^q > |S_\tau(\beta^*)| \tau^q. \end{align*} In both cases, we have the non-strict estimate \begin{align*} |S_\tau(\beta^*)| \tau^q \leq \sum_{j \in S_\tau(\beta^*)} |\beta_j^*|^q. \end{align*} [/step]

[step:Use the weak sparsity bound and rearrange] Since $S_\tau(\beta^*) \subseteq \{1,\dots,p\}$ and every term $|\beta_j^*|^q$ is non-negative, \begin{align*} \sum_{j \in S_\tau(\beta^*)} |\beta_j^*|^q \leq \sum_{j=1}^{p} |\beta_j^*|^q. \end{align*} Because $\beta^* \in \mathcal{B}_q(R)$, the defining inequality for $\mathcal{B}_q(R)$ gives \begin{align*} \sum_{j=1}^{p} |\beta_j^*|^q \leq R. \end{align*} Combining the preceding inequalities, \begin{align*} |S_\tau(\beta^*)| \tau^q \leq R. \end{align*} Since $\tau^q > 0$, division by $\tau^q$ yields \begin{align*} |S_\tau(\beta^*)| \leq R \tau^{-q}. \end{align*} This is the desired effective support bound. [/step]