How do we conduct the infimum of $inf_y()+⟨∇(),−⟩+\frac{L}{2}‖−‖^2$ in the quadratic upper bound

Question

I am trying to understand the Consequence of quadratic upper bound in the 16th slide of UCLA ese236c. The left-hand side of the inequity is still confusing to me. I know in this similar question, someone gave an explanation, but he concluded the infimum in one step without explanation. answer from another question How do we get the infimum?