We consider a convex function $f \in \mathcal{C}(\mathbb{R})$, admitting a minimizer $x_* \in \mathbb{R}$, and $\tau > 0$. The operator $p_f$ is defined as the unique minimizer of $F_{x_0}(x) := \frac{1}{2}|x - x_0|^2 + \tau f(x)$.\\
Show that $x_0 \in \mathbb{R}$ is a minimizer of $f$ if and only if $p_f(x_0) = x_0$.\\
Hint: consider the quantity $F_{x_0}((1-t)x_0 + tx_*)$ when $t \rightarrow 0^+$.