I am trying to solve the following problem: Given $\tau \in \mathbb{R}_{++}, \rho \in \mathbb{R}_{++},~y_n \in \mathbb{R}^n,~z_n \in \mathbb{R}^n$. Derive $$\text{prox}_{\frac{1}{\tau} g^{*}} (y_n + \frac{z_n}{\tau})$$, where $g=\rho\|\cdot\|$, and $\|\cdot\|$ denote the Euclidean norm.
Here are my attempt: $$\text{prox}_{\frac{1}{\tau} g^{*}} (y_n + \frac{z_n}{\tau})=\text{prox}_{\frac{1}{\tau} g^{*}}(\frac{y_n \tau + z_n}{\tau})$$. Let $u_n=y_n \tau + z_n$. By Moreau decomposition, for any $\lambda >0$, $x=\text{prox}_{\lambda f}(x)+\lambda\text{prox}_{\frac{1}{\lambda}f^*}(\frac{x}{\lambda})$. So $$\text{prox}_{\frac{1}{\tau} g^{*}} (\frac{u_n}{\tau})= \frac{ u_n- \text{prox}_{\tau g} (u_n) }{\tau}$$, where $$\text{prox}_{\tau g}(u_n)=\text{prox}_{\tau \|\cdot\|}(u_n)=\begin{cases} u_n-\frac{\tau u_n}{\|u_n\|}, \quad &\text{if}~ \|u_n\|>\tau\\ 0, \quad &\text{if $\|u_n\| \leq \tau $} \end{cases}$$. Hence $$ \text{prox}_{\frac{1}{\tau} g^{*}} (\frac{u_n}{\tau})= \frac{1}{\tau}(u_n- \text{prox}_{\tau g} (u_n))=\begin{cases} \frac{u_n}{\|u_n\|}, \quad &\text{if}~ \|u_n\|>\tau\\ \frac{u_n}{\tau}, \quad &\text{if $\|u_n\| \leq \tau $} \end{cases}$$ I am not sure if I did it correctly. I hope you can help me point out my mistakes if possible. Thank you
