How to prove the property of Lipschitz with regards to the Hessian matrix?

Question

Function $f$ is convex and twice differentiable, $\nabla f$ is Lipschitz with constant $L$.

How to prove that $\nabla^2 f(x) \preceq LI$ for all $x$？

Answer 1

Hint: $$\nabla^2 f(x) \, v = \lim_{t \searrow0} \frac{\nabla f(x+t \, v) - f(x)}{t}.$$