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 $x_{n+1} := p_f(x_n)$.\\
Show that for all $M, N \in \mathbb{N}$
$$|x_N - x_M| \leq \sqrt{2\tau|N-M|}\sqrt{\left|f(x_M) - f(x_N)\right|}$$