I am reading this paper: https://arxiv.org/pdf/1905.00313.pdf which is about the Polyak step size.
It states that in the proof of Lemma 2 (Second point) we use equation 2:
For smooth functions, equation (2) implies: $$d^2_{t+1}-d_t^2 \leq - \frac{\gamma h_t}{2\beta}$$
where $d_t = ||x_t-x^*||_2^2$ and $h_t = f(x_t)-f(x^*)$
When I look at equation 2, then I can follow the argument, assuming that $\frac{1}{2\beta} ||\nabla f(x_t)||^2 \leq h_t$ where $\beta$ is the smoothness parameter. This was proven by Lemma 4 in the "Elementary properties of convex analysis" appendix. However, I do not follow the proof for case 3 in for this Lemma. It starts with an assumption: $h_t \geq \frac{1}{\beta}||\nabla f(x_t)||^2$ which I do not understand.
Why are we able to say that for any smooth convex function, this holds: $h_t \geq \frac{1}{\beta}||\nabla f(x_t)||^2$ ? I assume that this is a fairly important statement, but can't find any reference in Boyds convex optimization.