We consider a convex function $f \in \mathcal{C}(\mathbb{R})$, admitting a minimizer $x_* \in \mathbb{R}$, and $\tau > 0$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := p_f(x_n)$. We have established that $p_f(x_{**}) = x_{**}$ for some $x_{**} \in \mathbb{R}$.\\
Conclude that $x_{**}$ is a minimizer of $f$, and that $x_n \rightarrow x_{**}$ as $n \rightarrow \infty$.