Conjugate of an infimal convolution ($(f\square g{)}^{\ast }=f^{\ast }+g^{\ast }$)

For proper convex functions whose infimal convolution is well defined, $(f\square g{)}^{\ast }=f^{\ast }+g^{\ast }$. The proof separates a supremum over independent variables after writing $x=u+y$. No attainment of the infimum is needed.