How does Lipschitz Continuity of the gradient implies that the Hessian-Matrix minus Lipschitz-Constant is negativ semidefinit?

Question

I was reading following lecture notes https://www.stat.cmu.edu/~ryantibs/convexopt-F13/scribes/lec6.pdf In the proof the author says that if the gradient $\nabla f$ is Lipschitz-continuous than $H_f(x)-LI$ is negative semidefinite, where L is the Lipschitz-constant and $I$ is the unit matrix. This means $$ ||\nabla f(x)-\nabla f(y)|| \leq L ||x-y|| \implies \forall x: x^T(H_f(x)-LI)x \leq 0$$ I don't know why this is true. Can someone show a proof of it? Thanks