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 for all $n \in \mathbb{N}$, assuming $x_0 \neq x_*$,
$$f(x_{n+1}) \leq f(x_n) - \frac{\tau}{2}(2 - \tau L)\frac{\left|f(x_n) - f(x_*)\right|^2}{\left|x_0 - x_*\right|^2}$$
Hint: use question 2.c)