Consider the function
$$f(x) := \frac{1}{3}x^3 \quad \text{if } x \geq 0, \quad f(x) := 0 \quad \text{if } x < 0$$
and the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_{n+1} := x_n - \tau f'(x_n)$.\\
We suppose only $\tau > 0$. Show that for all $x_0 \in \mathbb{R}$, the sequence $\left(x_n\right)_{n \in \mathbb{N}}$ converges to a minimizer of $f$.