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Let ${C \subset \mathbb{R}^n}$ be closed, convex, and nonempty. How might one show that the proximal mapping of the indicator function of $C$ is in fact the projection operator on to $C$?

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Proximal operator of the indicator function $\iota_C$ is given by \begin{equation*} prox_{\iota_C}(z)= \begin{aligned} & \underset{x}{\text{argmin}} & & \frac{1}{2}\|x-z\|^2_2 + \iota_C(x) \\ \end{aligned} \end{equation*} which can be re-written as: \begin{equation*} prox_{\iota_C}(z)= \begin{aligned} & \underset{x \in C}{\text{argmin}} & & \frac{1}{2}\|x-z\|^2_2 \\ \end{aligned} \end{equation*} which is nothing but the projection of $z$ onto $C$.

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