# Minimizing sequence $\implies$ Palais–Smale sequence

Set $$F:\mathbb{R}^n\rightarrow \mathbb{R}$$ a $$C^2$$-function that is bounded from below. Set $$x_n$$ a minimizing sequence, i.e., $$F(x_n)\to \alpha = \inf F$$. I want to prove that under the assumption of $$\lVert D^2 F\rVert$$ is bounded, $$x_n$$ is a Palais–Smale sequence, i.e., $$F(x_n)\to \alpha = \inf F$$ and $$\lVert DF(x_n)\rVert\to 0$$.

I have come across the variational Ekeland principle, with which I showed that for every minimizing sequence $$x_n$$ there exists another minimizing sequence $$y_n$$ close to $$x_n$$ in the sense that $$\lVert y_n-x_n\rVert\to 0$$ and $$\lVert DF(y_n)\rVert\rightarrow 0$$. It seems that this could be useful, but I am stuck. Any help is welcome!

• One approach could be mean value theorem – R. N. Marley May 14 at 8:26