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!