Lipschitz Smoothness implying strong convexity

Question

I recently stumbled upon this question.

After playing around with the definition a bit I am just not able to see why L-Lipschitz-Smoothness implies the inequality mentioned in 1:

$$ f(x + y) \leq f(x) + y^\top \nabla f(x) + \frac{L}{2} \| y \|^2 $$

I would love some pointers if anyone has any.

Answer 1

This exact proof is contained in Bertsekas' Nonlinear Programming 2nd Edition Proposition A. 24 called the "Descent Lemma".