I am trying to find the proximal point of the function $ f \left( x \right) = {\left\| x \right\|}_{2} $ where $ x \in \mathbb{R}^{n} $
The Proximal Operator is defined as:
$$ \operatorname{prox}_{\alpha f} \left( y \right) = \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{2} $$
I have tried doing this by finding the gradient for the entire function and also by differentiating along each coordinate but neither seems to work.
The corresponding proximal mapping is $$x\mapsto \Big(1-\frac{\alpha}{\max\big\{\|x\|,\alpha\big\}}\Big)x;$$ for details, see Bauschke-Combettes' Convex Analysis and Monotone Operator Theory in Hilbert Spaces (2017, second edition, Springer) Example 24.20.