Dynamic inventory

Derive Lemma 3

Lemma 3:.

If ft+1is convex and ft+1(x) = -c for x < S, where S satisfies the equation (S) = (cP-(1-)c)/(cP+cH), then the following hold:

a)S minimizes Gt(y) over all real y.

b)The optimal base stock level in period t is also S.

c)ftis convex and ft(x) = -c for x < S.

Derive a, b, c?