We are given $f \in \mathcal{C}^1(\mathbb{R})$, convex, admitting a minimizer $x_* \in \mathbb{R}$, with $f'$ being $L$-Lipschitzian, and $0 < \tau < 2/L$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := x_n - \tau f'(x_n)$.\\
Show that $\frac{\tau}{2}(2 - \tau L)\sum_{0 \leq i < n}\left|f'(x_i)\right|^2 \leq \left(f(x_0) - f(x_n)\right)$ for all $n \in \mathbb{N}^*$. Deduce that $f'(x_n) \rightarrow 0$ when $n \rightarrow \infty$.