We are given $f \in \mathcal{C}^1(\mathbb{R})$ such that $f'$ is $L$-Lipschitzian, with $L > 0$, and we fix $\tau$ such that $0 < \tau \leq 2/L$. We further suppose that $f$ is $\alpha$-convex, with $\alpha > 0$, that is
$$g(x) := f(x) - \frac{1}{2}\alpha x^2 \quad \text{is a convex function on } \mathbb{R}$$
Justify that $f'(x) - \alpha x$ is an increasing function of $x \in \mathbb{R}$. Deduce that $\alpha \leq L$.