I’ll prove half of the problem and leave the other half to you.

Suppose is a convex function, i.e. x_1,\ldots,x_m)\in S,\ x_{m+1}\geq f(x_1,\ldots,x_m)\}" alt="T=\{(x_1,\ldots,x_m,x_{m+1})x_1,\ldots,x_m)\in S,\ x_{m+1}\geq f(x_1,\ldots,x_m)\}" /> is a convex subset of .

Let .

Note that

Then, by definition of a convex set, for all

Thus, for all ,

The other case is proved in a similar way.