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 $\frac{1}{2}|x_1 - x_0|^2 + \tau f(x_1) \leq \tau f(x_0)$. Deduce that for all integers $N > M \geq 0$
$$\frac{1}{2}\sum_{M < n \leq N}|x_n - x_{n-1}|^2 \leq \tau\left(f(x_M) - f(x_N)\right)$$
Deduce that $|x_{n+1} - x_n| \rightarrow 0$ as $n \rightarrow \infty$.