Co-coercivity of gradient

Question

If $f$ is convex with $dom$ $f$ $= R^{n}$ and $g(x) = x^{T}x - f(x)$ is convex,

how to prove the Co-coercivity of gradient?

$$(\nabla f(x) - \nabla f(y))^{T}(x - y) \geq 1/L \parallel \nabla f(x) - \nabla f(y) \parallel ^{2}_{2}$$

Answer 1

I find the answer in ucla lecture Page 1-16.